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2. 0.1. maplllrHB3a~aqH no reOMeTpHHTIJIaHHMeTpHHHSJUlTeJlbCTBO tHayKa~, MOCKB8 3. I.F. SharyginProblems in Plane GeometryMirPublishersMoscow 4. Translated from Russianby Leonid LevantFirst published 1988Revised from the 1986 Russian editionm17020'0000-304 21-88 2056(01)-88 ' Ha ane..uucIWM Jl.awlCePrinted In the Union of Soviet Soetalis; Republic.@ HSAateJIbCTBO 4lHayxatI'nasaaa peAaxD;HJI t!B8HKo-MaTeM8TB1fOOKOiiJlHTepaTypw, t986@ English translation,Mir Publishers, 1988ISBN 5-03-000180-8 5. ContentsPreface to the English Edition . 6Section t. Fundamental Geometrical Factsand Theorems. ComputationalProblems . . . 8Section 2. Selected Problems and Theoremsof Plane Geometry 65. Carnot'sTheorem 65. Ceva's andMenelaus' Theorems. Affine Problems70. Loci of Points 82.Triangles. A Triangle and aCircle 89. Quadrilaterals t t 7.Circles aod Tangents. Feuerbach'sTheorem t29. Combinationsof Figures. Displacementsin the Plane. Polygons t37. GeometricalInequalities. Problemson Extrema 147Answers, Hint!, Solutions tooSection t tooSection 2 . . . 214Appendix. Inversion 392 6. Preface to the English EditionThis is a translation from the revisededition of the Russian book which wasissued in 1982. It is- actually the first ina two-volume work on solving problems ingeometry, the second volume "Problems inSolid Geometry" having been published inEnglish first by Mir Publishers in ~986.Both volumes are designed for schoolchildrenand teachers.This volume contains over 600 problemsin plane geometry and" consists of twoparts. The first part contains rather simpleproblems to be solved in classes and athome. The second part also contains hintsand detailed solutions. Over ~OO new problemshave been added to the 1982 edition,the simpler problems in the first additionhaving been eliminated, and a number ofnew sections- (circles and tangents, polygons,combinations of figures, etc.) havingbeen introduced, The general structure ofthe book has been changed somewhat toaccord with the new, more detailed, classificationof the problems. As a result, allthe problems in this volume have beenrearranged.Although the problems in this collectionvary in "age" (some of them can be foundin old books and journals, others wereoffered at mathematical olympiads or publishedin the journal "Quant" (Moscow,I still hope that some of the problems in 7. Prefacethis collection will be of interest to experiencedgeometers.Almost every geometrical problem is nonstandard(as compared with routine exerciseson solving equations, inequalities,etc.): one has to think of what additionalconstructions must be made, or which formulasand theorems must be used. Therefore,this collection cannot be regarded asa problem-book in geometry; it is rather acollection of geometrical puzzles aimed atdemonstrating the elegance of elementarygeometrical techniques of proof and methodsof computation (without using vector algebraand with a minimal use of the methodof coordinates, geometrical transformstions,though a somewhat wider use of trigonometry).In conclusion, I should like to thankA.Z. Bershtein who assisted me in preparingthe first section of the book for print.I am also grateful to A.A. Yagubiants wholet me know several elegant geometricalfacts.The Author 8. Section 1Fundamental Geometrical Factsand Theorems.Computational Problemst. Prove that the medians in a triangleintersect at one point (the median point)and are divided by this point in the ratio1 : 2.2. Prove that the medians separate thetriangle into six equivalent parts.3. Prove that the diameter of the circlecircumscribed about a triangle is equalto the ratio of its side to the sine of theopposite angle.4. Let the vertex of an angle be locatedoutside a circle, and let the sides of theangle intersect the circle. Prove that theangle is measured by the half-difference ofthe arcs inside the angle which are cut outby its sides on the circle.5. Let the vertex of an angle lie insidea circle. Prove that the angle is measuredby the half-sum of the arcs one of whichis enclosed between its sides and the otherbetween their extensions.6. Let AB denote a chord of a circle, andl the tangent to the circle at the point A.Prove that either of the two angles betweenAB and l is measured by the half-arc of the 9. Sec. f. Fundamental Facts 9circle enclosed inside the angle under consideration.7. Through the point M located at a distancea from the centre of a circle of radius R(a > R), a secant is drawn intersecting thecircle at points A and B. Prove that theproduct IMA 11 MB J is constant for allthe secants and equals a2 - R2 (which isthe squared length of the tangent).8. A chord AB is drawn through the pointM situated at a distance a from the centreof a circle of radius R (a < R). Prove thatIAM I-I MB I is constant for all the chordsand equals RI - a29. Let AM be an angle bisector in thetriangle ABC. Prove that IBM I: I CMf =IAB I IAC I. The same is true forthe bisector of the exterior angle of thetriangle. (In this case the .point M lies on theextension of the side Be.)10. Prove that the sum of the squares ofthe lengths of the diagonals of a parallelogramis equal to the sum of the squares ofthe lengths of its sides.1t. Given the sides of a triangle (a, b,and e). Prove that the median mel drawn tothe side a can be computed by the formulama = ~ V2lJ2+ 2c2-a212. Given two triangles having one vertexA in common, the other vertices beingsituated on two straight lines passing 10. 10 Problems in Plane Geometrythrough A. Prove that the ratio of the areasof these triangles is equal to the ratio ofthe products of the two sides of each triangleemanating from the vertex A.13. Prove that the area of the circumscribedpolygon is equal to rp, where ris the radius of the inscribed circle and pits half-perimeter (in particular, this formulaholds true for a triangle).14. Prove that the area of a quadrilateralis equal to half the product of its diagonals'and the sine of the angle between them. '15.Prove the validity of the followingformulas for the area of a triangle:al sin B sin C 2... S = 2s.mA'S= 2R SID A SID B sm C,where A, B, C are its angles, a is the sidelying opposite the angle A, and R is theradius of the circumscribed circle.16. Prove that the radius of the circleinscribed in a right triangle can be com-a+b-c puted by the formula r = 2 'where a and b are the legs and c is thehypotenuse.17. Prove that if a and b are two sides ofa triangle, a the angle between them, and lthe bisector of this angle, then(2abc08 T1= a+b 11. Sec. 1. Fundamental Facts it18. Prove that the distances from thevertex A of the triangle ABC to the pointsof tangency of the inscribed circle with thesides AB and AC are equal to p - a (each),where p is the half-perimeter of the triangleABC, a = I BC I.19. Prove that if in a convex quadrilateralABeD I AB I + I CD I = I AD I +I BC I, then there is a circle touching allof its sides.20. (a) Prove that the altitudes in atriangle 'are concurrent (that is intersect atone point). (b) Prove that the distancefrom any vertex of a triangle to the pointof intersection of the altitudes is twicethe distance from the centre of the circumscribedcircle to the opposite side. 21. Points A and B are taken on one sideof a right angle with vertex 0 and lOA I =a, I OB I = b. Find the radius of the circlepassing through the points A and Bandtouching the other side of the angle.22. The hypotenuse of a right triangle is.equal to c, one of the acute angles being30. Find the radius of the circle withcentre at the vertex of the angle of 30which separates the triangle into two equi valentparts.23. The legs of a right triangle are a andb. Find the distance from the vertex of the 12. t2 Problems in Plane Geometryright angle to the nearest point of theinscribed circle.24. One of the medians of a right triangleis equal to m and divides the right anglein the ratio 1 : 2. Find the area of thetriangle.25. Given in a triangle ABC are threesides: I Be I = a, I CA I = b, I AB I = c.Find the ratio in which the point of intersectionof the angle bisectors divides thebisector of the angle B.26. Prove that the sum of the distancesfrom any point of the base of an isoscelestriangle to its sides is equal to the altitudedrawn to either of the sides.1:1. Prove that the sum of distances fromany point inside an equilateral triangleto its sides is equal to the altitude of thistriangle.28. In an isosceles triangle ABC, takenon the base AC is a point M such thatIAM I = a, IMe I = b. Circles are inscribedin the triangles ABM and CBM.Find the distance between the points atwhich these circles touch the side BM.29. Find the area of the quadrilateralbounded by the angle bisectors of a parallelogramwith sides a and b and angle Ct.30. A circle is inscribed in a rhombuswith altitude h and acute angle C%. Findthe radius of the greatest of two possiblecircles each of which touches the givencircle and two sides of the rhombus. 13. Sec. 1. Fundamental Facts 133t . Determine the acute angle of therhombus whose side is the geometric meanof its diagonals.32. The diagonals of a convex quadrilateralare equal to a and b, the line segmentsjoining the midpoints of the opposite sidesare congruent. Find the area of the quadrilateral.33. The side AD of the rectangle ABCDis three times the side AB; points M andN divide AD into three equal parts. FindLAMB + LANB + LADB.34. Two circles intersect at points Aand B. Chords A C and AD touching thegiven circles are drawn through the pointA. Prove that lAC 12.1 BD I = IAD 21IBC I.35. Prove that the bisector of the rightangle in a right triangle bisects the anglebetween the median and the altitude drawnto the hypotenuse.36. On a circle of radius r, three pointsare chosen 80 that the circle is dividedinto three arcs in the ratio 3 : 4 : 5. At thedivision points, tangents are drawn to thecircle. Find the area of the triangle formedby the tangents.37. An equilateral trapezoid is circumscribedabout a circle, the lateral side ofthe trapezoid is I, one of its bases is equalto 4. Find the area of the trapezoid.38. Two straight lines parallel to thebases of a trapezoid divide each lateral 14. 14 Problems in Plane Geometryside into three equal parts. The entiretrapezoid is separated by the lines intothree parts. Find the area of the middlepart if the areas of the upper and lowerparts are 81 and S2' respectively.39. In the trapezoid ABCD I AB I = a,IBC I = b (a =1= b). The bisector of theangle A intersects either the base BC or thelateral side CD. Find out which of them?40. Find the length of the line segmentparallel to the bases of a trapezoid andpassing through the point of intersectionof its diagonals if the bases of the trapezoidare a and b.41. In an equilateral trapezoid circumscribedabout 8 circle, the ratio of theparallel sides is k. Find the angle at thebase.42. In a trapezoid ABCD, the base ABis equal to a, and the base CD to b. .Findthe area of the trapezoid if the diagonalsof the trapezoid are known to be the bisectorsof the angles DAB and ABC.43. In an equilateral trapezoid, the midlineis equal to a, and the diagonals aremutually perpendicular. Find the area ofthe trapezoid.44. The area of an equilateral trapezoidcircumscribed about a circle is equal to S,and the altitude of the trapezoid is halfits lateral side. Determine the radius ofthe circle inscribed in the trapezoid.45. The areas of the triangles formed by 15. Sec. f. Fundamental Facts i5the segments of the diagonals of a trapezoidand its bases are equal to 81 and 8 1 Findthe area of the trapezoid.46. In a triangle ABC, the angle ABC istX. Find the angle AOe, where 0 is thecentre of the inscribed circle.47. The bisector of the right angle isdrawn in a right triangle. Find the distancebetween the points of intersection of thealtitudes of the triangles thus obtained,if the legs of the given triangle are a and b.48. A straight line perpendicular to twosides of a parallelogram divides the latterinto two trapezoids in each of which acircle can .be inscribed. Find the acuteangle of the parallelogram if its sides area and b (a < b). .49. Given a half-disc with diameter AB.Two straight lines are drawn through themidpoint of the semicircle which dividethe half-disc into three equivalent areas.In what ratio is the diameter AB dividedby these lines?50. A square ABCD with side a and twocircles are constructed. The first circle isentirely inside the square touching the sideAB at a point E and also the side Be anddiagonal AC. The second circle with centreat A passes through the point E. Findthe area of the common part of the twodiscs bounded by these circles.51. The vertices of a regular hexagonwith side a are the centres of the circles 16. 16 Problems in Plane Geometrywith radius a/y'2. Find the area of thepart of the hexagon not enclosed by thesecircles.52. A point A is taken outslde a circleof radius R. Two secants are drawn fromthis point: one passes through the centre,the other at a distance of R/2 from thecentre. Find the area of the region enclosedbetween these secants.53. In a quadrilateral ABeD: LDAB =90, LDBC = 90. IDB I= a, and IDC 1=b. Find the distance between the centresof two circles one of which passes throughthe points D, A and B, the other throughthe points B, C, and D.54. On the sides AB and AD of therhombus ABCD points M and N are takensuch that the straight lines Me and NCseparate the rhombus into three equivalentparts. Find IMN I if I BD I = d.55. Points M and N are taken on theside AB of a triangle ABC such thatlAM I: IMN I: INB I = 1: 2 : 3.',Throughthe points M and N straight Iines aredrawn parallel to tHe side AC. Find thearea of the part of the triangle enclosedbetween these lines if the area of the triangleABC is equal to S.56. Given a circle and a point A locatedoutside of this circle, straight lines ABand AC are tangent to it (B and C pointsof tangency). Prove that the centre of the 17. Sec. t. Fundamental Facts i7circle inscribed in the triangle ABC lieson the given circle.57. A circle is circumscribed about anequilateral triangle ABC, and an arbitrarypoint M is taken on the arc BC. Prove thatIAM I = I EM I + I CM I58. Let H be the point of intersection ofthe altitudes in a triangle ABC. Find theinterior angles of the triangle ABC ifLBAH = a, LABH = ~.59. The area of a rhombus is equal to S,the sum of its diagonals is m, Find the sideof the rhombus.60. A square with side a is inscribed ina circle. Find the side of the square inscribedin one of the segments thus obtained.61. In a 1200 segment of a circle withaltitude h a rectangle ABCD is inscribedso that I AB I : I BC I = 1 4 (BC lieson the chord). Find the area of the rectangle.62. The area of an annulus is equal to S.The radius of the larger circle is equal tothe circumference of the smaller. Find theradius of the smaller circle.63. Express the side of a regular decagonin terms of the radius R of the circumscribedcircle.64. Tangents MA and MB are drawnfrom an exterior point M to a circle ofradius R forming an angle a. Determine2-01557 18. 18 Problems in Plane Geometrythe area of the figure bounded by the tangentsand the minor arc of the circle.65. Given a square ABCD with side a.Find the centre of the circle passingthrough the following points: the midpointof the side AB, the centre of the square,and the vertex C.66. Given a rhombus with side a and acuteangle Ct. Find the radius of the circle passingthrough two neighbouring vertices ofthe rhombus and touching the oppositeside of the rhombus or its extension.67. Given three pairwise tangent circlesof radius r. Find the area of the triangleformed by three lines each of which touchestwo circles and does not intersect the thirdone.68. A circle of radius r touches a straightline at a point M. Two, points A and Bare chosen on this line on opposite sides ofM such that I MAl = 1MB I = a. Findthe radius of the circle passing through Aand B and touching the given circle.69. Given a square ABCD with side a.Taken on the side BC is a point M such thatIBM I = 3 I MC I and on the side CD apoint N such that 2 I CN I = I ND I. Findthe radius of the circle inscribed in thetriangle AMN.70. Given a square ABeD with side a.Determine the distance between the midpointof the line segment AM, where M is 19. Sec. 1. Fundamental Facts 19the midpoint of Be, and a point N on theside CD such that I CN I I ND I = 3 1.71. A straight line emanating from thevertex A in a triangle ABC bisects themedian BD (the point D lies on the sideAC). What is the ratio in which this linedivides the side BC?72. In a right triangle ABC the leg CAis equal to b, the leg eB is equal to a, CHis the altitude, and AM is the median.Find the area of the triangle BMH.73. Given an isosceles triangle ABC whoseLA = a > 90 and I Be I = a. Find thedistance between the point of intersectionof the altitudes and the centre of the circumscribedcircle.74. A circle is circumscribed about atriangle ABC where I BC I = a, LB = a,LC = p. The bisector of the angle A meetsthe circle at a point K. Find IAK I.75. In a circle of radius R, a diameter isdrawn with a point A taken at a distancea from the centre. Find the radius of anothercircle which is tangent to the diameter atthe point A and touches internally thegiven circle.76. In a circle, three pairwise intersectingchords are drawn. Each chord is dividedinto three equal parts by the points ofintersection. Find the radius of the circleif one of the chords is equal to a.77. One regular hexagon is inscribed ina circle, the other is circumscribed about2* 20. 20 Problems in Plane Geometryit. Find the radius of the circle if the differencebetween the perimeters of these hexagonsis equal to a.78. In an equilateral triangle ABC whoseside is equal to a, the altitude BK is drawn.A circle is inscribed in each of the trianglesABK and BCK, and a common externaltangent, different from the side AC, is drawnto them. Find the area of the triangle cutoff by this tangent from the triangle ABC.79. Given in an inscribed quadrilateralABCD are the angles: LDAB = a, LABC=p, LBKC = y, where K is the pointof intersection of the diagonals. Find theangle ACD.80. In an inscribed quadrilateral ABeDwhose diagonals intersect at a point K,lAB I = a, 18K 1= b, IAK 1= c, I CDI=d. Find I AC I.8t. A circle is circumscribed about atrapezoid. The angle between one of thebases of the trapezoid and a lateral side isequal to ex and the angle between this baseand one of the diagonals is equal to p.Find the ratio of the area of the circle tothe area of the trapezoid.82. In an equilateral trapezoid ABCD,the base AD is equal to a, the base Beis equal to b, I AB I = d. Drawn throughthe vertex B is a straight line bisecting thediagonal AC and intersecting AD at a pointK. Find the area of the triangle BDK .83. Find the sum of the squares of the 21. Sec. 1. Fundamental Facts 21distances from the point M taken on a diameterof a circle to the end points of anychord parallel to this diameter if the radiusof the circle is R, and the distance from Mto the centre of the circle is a.84. A common chord of two intersectingcircles can be observed from their centresat angles of 90 and 60. Find the radii ofthe circles if the distance between theircentres is equal to a.85. Given a regular triangle ABC. A pointK di vides the side A C in the ratio 2 : 1,and a point M divides the side AB in theratio t 2 (as measured from the vertex Ain both cases). Prove that the length of theline segment KM is equal to the radius ofthe circle circumscribed about the triangleABC.86. Two circles of radii Rand R/2 toucheach other externally. One of the end pointsof the line segment of length 2R formingan angle of 30 with the centre line coincideswith the centre of the circle of the smallerradius. What part of the line segment liesoutside both circles? (The line segmentintersects both circles.)87. A median BK, an angle bisector BE,and an altitude AD are drawn in a triangleABC. Find the side AC if it is known thatthelines EX and BE divide the line segmentAD into three equal parts and IAB I = 4.88. The ratio of the radius of the circleinscribed in an isosceles trrangle to the 22. 22 Problems in Plane Geometryradius of the circle circumscribed about thistriangle is equal to k. Find the base angleof the triangle.89. Find the cosine of the angle at thebase of an isosceles triangle if the point ofintersection of its altitudes lies on the circleinscribed in the triangle.90. Find the area of the pentagon boundedby the lines BC, CD, AN, AM, and BD,where A, B, and D are the vertices of asquare ABCD, N the midpoint of the sideBC, and M divides the side CD in the ratio2 : 1 (counting from the vertex C) if theside of the square ABCD is equal to a.91. Given in a triangle ABC: LBAC =a, LABC =~. A circle centred at Bpasses through A and intersects the line ACat a point K different from A, and the lineBe at points E and F. Find the angles ofthe triangle EKF.92. Given a square with side a. Find thearea of the regular triangle one of whosevertices coincides with the midpoint of oneof the sides of the square, the other twolying on the diagonals of the square.93. Points M, N, and K are taken on thesides of a square ABeD, where M is themidpoint of AB, N lies on the side BC(2 I BN I = I NC I). K lies on the sideDA (2 IDK I = I KA I). Find the sineof the angle between the lines MC and N K.94. A circle of radius r passes through thevertices A and B of the triangle ABC and 23. Sec. 1. Fundamental Facts 23intersects the side BC at a point D. Findthe radius of the circle passing through thepoints A,D, and C if IAB 1=c, IAC I=b.95. In a triangle ABC, the side AB isequal to 3, and the altitude CD droppedon the side AB is equal to V3: The foot Dof the altitude CD lies on the side AB, andthe line segment AD is equal to the side BC.Find I AC I.96. A regular hexagon ABCDEF is inscribedin a circle of radius R. Find theradius of the circle inscribed in the triangleACD.97. The side AB of a square ABCD isequal to 1 and is a chord of a circle, therest of the sides of the square lying outsidethis circle. The length of the tangent CKdrawn from the vertex C to the circle isequal to 2. Find the diameter of the circle.98. In a right triangle, the smaller angleis equal to a. A straight line drawn perpendicularlyto the hypotenuse dividesthe triangle into two equivalent parts.Determine the ratio in which this linedivides the hypotenuse.99. Drawn inside a regular triangle withside equal to 1 are two circles touchingeach other. Each of the circles touches twosides of the triangle (each side of thetriangle touches at least one of the circles).Prove that the sum of the radii of thesecircles is not less than (va - 1}/2. 24. 24 Problems in Plane GeometrytOO. In a right triangle ABC with anacute angle A equal to 30, the bisector ofthe other acute angle is drawn. Find thedistance between the centres of the twocircles inscribed in the triangles ABD andCBD if the smaller leg is equal to t.10t. In a trapezoid ABeD, the anglesA and D at the base AD are equal to 60and 30, respectively. A point N lies onthe base Be, and IBN I : I NC I = 2.A point M lies on the base AD; the straightline M N is perpendicular to the bases ofthe trapezoid and divides its area intotwo equal parts. Find IAM I : IMD I.102. Given in a triangle ABC: I Be I =a, LA = a, LB = p. Find the radius ofthe circle touching both the side ACata point A and the side BC.103. Given in a triangle ABC: I AB I =c, IRG I = a, LB =~. On the side AR,a point M is taken such that 2 IAM I =3 IMB I. Find the distance from M to themidpoint of the side AC. t04. In a triangle ABC, a point M is takenon the side AB and a point N on the sideAC such that IAM I = 3 I MB I and2 IAN I = I NC I. Find the area of thequadrilateral MBCN if the area of thetriangle ABC is equal to S.t05. Given two concentric circles .ofradii Rand r (R > r) with a common centreO. A third circle touches both of them.Find the tangent of the angle between the 25. Sec. t. Fundamental Facts 25tangent lines to the third circle emanatingfrom the point O.t06. Given in a parallelogram ABeD:I AB I = a, IAD I = b (b > a), LBAD =a (ex < 90). On the sides AD and BC,points K and M are taken such that BKDMis a rhombus. Find the side of the rhombus.107. In a right triangle, the hypotenuseis equal to c. The centres of three circlesof radius cl5 are found at its vertices. Findthe radius of a fourth circle which touchesthe three given circles and does not enclosethem.108. Find the radius of the circle whichcuts on both sides of an angle ex chords oflength a if the distance between the nearestend points of these chords is known to beequal to b.109. A circle is constructed on the sideBC of a triangle ABC as diameter. Thiscircle intersects the sides AB and AC atpoints M and N, a' .respectively. Find thearea of the triangle AMN if the area of thetriangle ABC is equal to S, and LBAC=a.110. In a circle of radius R two mutuallyperpendicular chords MNand PQ aredrawn. Find the distance between the pointsM and P if I NQ I = a.ttl. In a triangle ABC, on the largestside BC equal to b, a point M is chosen.Find the shortest distance between thecentres of the circles circumscribed aboutthe triangles BAM and AGM. 26. 26 Problems in Plane Geometry112. Given in a parallelogram ABeD:IAB I = a, I Be I = b, LABC = ct. Findthe distance between the centres of thecircles circumscribed about the trianglesBCD and DAB.113. In a triangle ABC, LA = a, I BA 1=a, lAC 1 = b. On the sides AC andAR, points M and N are taken, M being themidpoint of AC. Find the length of theline segment MN if the area of the triangleAMN is 1/3 of the area of the triangle ABC.114. Find the angles of a rhombus if thearea of the circle inscribed in it is halfthe area of the rhombus.tt5. Find the common area of two equalsquares of side a if one can be obtainedfrom the other by rotating through an angleof 45 about its vertex.116. In a quadrilateral inscribed in acircle, two opposite sides are mutuallyperpendicular, one of them being equal toa, the adjacent acute angle is divided byone of the diagonals into ct and ~. Determinethe diagonals of the quadrilateral (theangle a is adjacent to the given side).117. Given a parallelogram ABeD withan acute angle DAB equal to a in whichIAB I = a, IAD I = b (a < b). Let Kdenote the foot of the perpendicular droppedfrom the vertex B on AD, and M the footof the perpendicular dropped from the pointK on the extension of the side CD. Find thearea of the triangle BKM. 27. Sec. 1. Fundamental Facts 27118. In a triangle ABC, drawn from thevertex C are two rays dividing the angleACB into three equal parts. Find the ratioof the segments of these rays enclosedinside the triangle if I BC I = 3 I AC I,LACB = Ct.119. In an isosceles triangle ABC (I AB 1=I BC I) the angle bisector AD is drawn.The areas of the triangles ABD and ADCare equal to 8 1 and 8 2 , respectively. FindlAC I.120. A circle of radius R1 is inscribedin an angle Ct. Another circle of radius R 2touches one of the sides of the angle atthe same point as the first one and intersectsthe other side of the angle at pointsA and B. Find IAB I.121. On a straight line passing throughthe centre 0 of the circle of radius 12,points A and B are taken such that 1OA I =15, I AB 1 = 5. From the points A and B,tangents are drawn to the circle whose pointsof tangency lie on one side of the lineOAB. Find the area of the triangle ABC,where C is the point of intersection of thesetangents.122. Given in a triangle ABC: I BC I =a, LA = a, LB = p. Find the radiusof the circle intersecting all of i ts sidesand cutting off on each of them a chord oflength d.123. In a convex quadrilateral, the linesegments joining the midpoints of the oppo- 28. 28 Problems in Plane Geometrysite sides are equal to a and b and intersectat an angle of 60 Find the diagonals ofthe quadrilateral.124. In a triangle ABC, taken on theside Be is a point M such that the distancefrom the vertex B to the centre of gravityof the triangle AMC is equal to the distancefrom the vertex C to the centre of gravityof the triangle AMB. Prove that IBM I =IDC I where D is the foot of the altitudedropped from the vertex A to Be.125. In a right triangle ABC, the bisectorBE of the right angle B is divided by thecentre 0 of the inscribed circle so thatI BO I I OE I = V3 V2. Find the acuteangles of the triangle.126. A circle is constructed on a linesegment AB of length R as diameter. A secondcircle of the same radius is centred atthe point A. A third circle touches thefirst circle internally and the second circleexternally; it also touches the line segmentAB. Find the radius of the third circle.127. Given a triangle ABC. It is knownthat IAB I = 4, I AC I = 2, and I BC I =3. The bisector of the angle A intersectsthe side BC at a point K. The straight linepassing through the point B and beingparallel to AC intersects the extension ofthe angle bisector AK at the point M. FindIKMI128. A circle centred inside a right angletouches one of the sides of the angle, inter- 29. Sec. 1. Fundamental Facts 29sects the other side at points A and Bandintersects the bisector of the angle at pointsC and D. The chord AB is equal to V~the chord CD to V7. Find the radius ofthe circle.129. Two circles of radius 1 lie in a parallelogram,each circle touching the othercircle and three sides of the parallelogram.One of the segments of the side from thevertex to the point of tangency is equalto Va Find the area of the parallelogram.130. A circle of radius R passes throughthe vertices A and B of the triangle ABCand touches the line AC at A. Find thearea of the triangle ABC if LB = a, LA =~.131. In a triangle ABC, the angle bisectorAK is perpendicular to the median BM,and the angle B is equal to 120. Find theratio of the area of the triangle A-BC to thearea of the circle circumscribed about thistriangle.132. In a right triangle ABC, a circletouching the side Be is drawn through themidpoints of AB and AC. Find the part ofthe hypotenuse AC which lies inside thiscircle if I AB I = 3, I BC I = 4.133. Given a line segment a. Threecircles of radius R are centred at the endpoints and midpoint of the line segment.Find the radius of the fourth circle whichtouches the three given circles. 30. 30 Problems in Plane Geometry134. Find the angle between the commonexternal and internal tangents to two circlesof radii Rand r if the distance betweentheir centres equals -V 2 (R2 + r 2) (the centresof the circles are on the same side of thecommon external tangent and on both sidesof the common internal tangent).135. The line segment AB is the diameterof a circle, and the point C lies outsidethis circle. The line segments AC and BCintersect the circle at points D and E,respectively. Find the angle CBD if theratio of the areas of the triangles DCE andABC is 1 4.136. In a rhombus ABCD of side a, theangle at the vertex A is equal to 1200Points E and F lie on the sides BC and AD,respectively, the line segment EF and thediagonal AC of the rhombus intersect at M.The ratio of the areas of the quadrilateralsREFA and ECDF is 1 : 2. Find IEM Iif IAM I I MC I = 1 3.137. Given a circle of radius R centredat O. A tangent AK is drawn to the circlefrom the end point A of the line segmentVA, which meets the circle at M. Findthe radius of the circle touching the linesegments AK, AM, and the arc MK ifLOAK = 60.t38. Inscribed in a circle is an isoscelestriangle ABC in which I AB I = I Be Iand LB = p. The midline of the triangle 31. Sec. 1. Fundamental Facts 31is extended to intersect the circle at pointsD and E (DE II AC). Find the ratio of theareas of the triangles ABC and pBE.t39. Given an angle ex with vertex O.A point M is taken on one of its sides anda perpendicular is erected at this pointto intersect the other side of the angle at apoint N. Just in the same way, at a pointK taken on the other side of the angle aperpendicular is erected to intersect thefirst side at a point P. Let B denote thepoint of intersection of the lines MNandKP, and A the point of intersection of thelines OB and NB. Find lOA I if 10M I =a and I OP I === b.t40. Two circles of radii Rand r touchthe sides of a given angle and each other.Find the radius of a third circle touchingthe sides of the same angle and whose centreis found at the point at which the givencircles touch each other.t41. The distance between the centresof two non-intersecting circles is equal to a.Prove that the four points of intersectionof common external and internal tangentslie on one circle. Find the radius of thiscircle.t42. Prove that the segment of a commonexternal tangent to two circles which isenclosed between common internal tangentsis equal to the length of a common internaltangent.143. Two mutually perpendicular ra- 32. 32 Problems in Plane Geometrydii VA and OB are drawn in a circlecentred at O. A point C is on the arc ABsuch that LAOe = 60 (LBOC = 30).A circle of radius AB centred at A intersectsthe extension of OC beyond the pointC at D. Prove that the line segment CDis equal to the side of a regular decagoninscribed in the circle.Let us now take a point M diametricallyopposite to the point C. The line segmentMD, increased by 1/5 of its length, is assumedto be approximately equal to half thecircumference. Estimate the error of thisapproximation.144. Given a rectangle 7 X 8. One vertexof a regular triangle coincides with one ofthe vertices of the rectangle, the two othervertices lying on its sides not containingthis vertex. Find the side of the regulartriangle.145. Find the radius of the minimal circlecontaining an equilateral trapezoid withbases of 15 and 4 and lateral side of 9..146. ABCD is a rectangle in whichI AB I = 9, I BC I = 7. A point M istaken on the side CD such that I CM I =3, and point N on the side AD such thatI AN I = 2.5. Find the greatest radius ofthe circle which goes inside the pentagonABCMN.147. Find the greatest angle of a triangleif the radius of the circle inscribed in thetriangle with vertices at the feet of the 33. Sec. 1. Fundamental Facts 33altitudes of the given triangle is half theleast altitude of the given triangle.148. In a triangle ABC, the bisector ofthe angle C is perpendicular to the medianemanating from the vertex B. The centreof the inscribed circle lies on the circlepassing through the points A and C and thecentre of the circumscribed circle. FindIAB I if I BC I = 1.149. A point M is at distances of 2, 3 and6 from the sides of a regular triangle (thatis, from the lines on which its sides aresituated). Find the side of the regulartriangle if its area is less than 14.150. A point M is at distances -of vaand 3 va from the sides of an angle of 60(the feet of the perpendiculars dropped fromM on the sides of the angle lie on the sidesthemselves, but not on their extensions).A straight line passing through the point Mintersects the sides of the angle and cuts offa triangle whose perimeter is 12. Find thearea of this triangle.151. Given a rectangle ABeD in whichI AB I = 4, I Be I = 3. Find the side ofthe rhombus one vertex of which coincideswith A, and three others lie on the linesegments AB, BC and BD (one vertex oneach segment).152. Given a square ABCD with a sideequal to 1. Find the side of the rhombus onevertex of which coincides with A, the oppo-3-01557 34. 34 'Problems in Plane Geometrysite vertex lies on the line BD, and the tworemaining vertices on the lines BC and CD.153. In a parallelogram ABCD the acuteangle is equal to cx,. A circle of radius rpasses through the vertices A, B, and Cand intersects the lines AD and CD at pointsM and N. Find the area of the triangleBMN.154. A circle passing through the verticesA, B, and C of the parallelogram ABCDintersects the lines AD and CD at pointsM and N. The point M is at distances of4, 3 and 2 from the vertices B, C, and D,respectively. Find I MN I.155. Given a triangle ABC in whichLBAC = n/6. The circle centred at Awith radius equal to the altitude droppedon BC separates the triangle into two equalareas. Find the greatest angle of the triangleABC.156. In an isosceles triangle ABC LB =120. Find the common chord of two circles:one is circumscribed about ABC, the otherpasses through the centre of the inscribedcircle and the feet of the bisectors of theangles A and C if I AC I = 1.157. In a triangle ABC the side Be isequal to a, the radius of the inscribed circleis equal to r, Determine the radii of twoequal circles tangent to each other, one ofthem touching the sides Be and BA, theother-the sides Be and CA.158. A trapezoid is inscribed in a circle 35. Sec. 1. Fundamental Facts. 35of radius R. Straight lines passing throughthe end points of one of the bases of thetrapezoid parallel to the lateral sides intersectat the centre of the circle. The lateralside .can be observed from the centre at anangle tX. Find the area of the trapezoid.159. The hypotenuse of a right triangleis equal to c. What are the limits of changeof the distance between the centre of theinscribed circle and the point of intersectionof the medians?160. The sides of a parallelogram areequal to a and b (a =1= b). What are thelimits of change of the cosine of the acuteangle between the diagonals?161. Three straight lines are drawnthrough a point M inside a triangle ABCparallel to its sides. The segments of thelines enclosed inside the triangle are equalto one another. Find their length if thesides of the triangle are a, b, and c.162. Three equal circles are drawn insidea triangle ABC each of which touches twoof its sides. The three circles have a commonpoint. Find their radii if the radii of thecircles inscribed in and circumscribed aboutthe triangle ABe are equal to rand R,respectively.163. In a triangle ABC, a median ADis drawn, LDAC + LABC = 90 FindLBA"C if I AB I =1= IAC I164. Three circles of radii 1, 2, and 3touch one another externally. Find the3* 36. 36 Problems in Plane Geometryradius of the circle passing through thepoints of tangency of these circles.165. A square of unit area is inscribedin an isosceles triangle, one of the sidesof the square lies on the base of the triangle.Find the area of the triangle if the centresof gravity of the triangle and square areknown to coincide.166. In an equilateral triangle ABC,the side is equal to a. Taken on the side Beis a point D, and on the side AB a point Esuch that 1BD 1 = a/3, 1AE 1 = I DE I.Find 1 CE I167. Given a right triangle ABC. Theangle bisector CL (I CL I = a) and themedian cu (I CM I = b) are drawn fromthe vertex of the right angle C. Find thearea of the triangle ABC.168. A circle is inscribed in a trapezoid.Find the area of the trapezoid given thelength a of one of the bases and the linesegments band d into which one of thelateral sides is divided by the point oftangency (the segment b adjoins the basea).169. The diagonals of a trapezoid are equalto 3 and 5, and the line segment joining themidpoints of the bases is equal to 2. Findthe area of the trapezoid.170. A circle of radius 1 is inscribed in atriangle ABC for which cos B = 0.8. Thiscircle touches the midline of the triangleABC parallel to the side AC. Find AC. 37. Sec. 1. Fundamental Facts 37171. Given a regular triangle ABC ofarea S. Drawn parallel to its sides at equaldistances from them are three straight linesintersecting inside the triangle to form atriangle AIBICl whose area is Q. Find thedistance between the parallel sides of thetriangles ABC and AIBICl 172. The sides AB and CD of a quadrilateralABeD are mutually perpendicular;they are the diameters of two equal circlesof radius r which touch each other. Findthe area of the quadrilateral ABCD ifI Be I : IAD I = k.173. Two circles touching each other areinscribed in an angle whose size is ct.Determine the ratio of the radius of thesmaller circle to the radius of a third circletouching both the circles and one of thesides of the angle.174. In a triangle ABC, circle intersectingthe sides AC and BC at points M andN, respectively, is constructed on theMidline DE, parallel to AB, as on thediameter. Find IMN I if IBC I = a, lAC I=b, I AB 1= c.175. The distance between the centresof two circles is equal to a. Find the side ofa rhombus two opposite vertices of whichlie on one circle, and the other two on theother if the radii of the circles are Rand r.176. Find the area of the rhombus LiBeDif the radii of the circles circumscribed 38. Problems in Plane Geometryabout the triangles ABC and ABD are Rand r, respectively.177. Given an angle of size a with vertexat A and a point B at distances a and bfrom the sides of the angle. Find I AB I.t 78. In a triangle ABC, the altitudes haand hb drawn from the vertices A and B,respectively, and the length l of the bisectorof the angle C are given. Find Le.179. A circle is circumscribed about aright triangle. Another circle of the sameradius touches the legs of this triangle, oneof the vertices of the triangle being one ofthe points of tangency. Find the ratioof the area of the triangle to the area ofthe common part of the two given circles.180. Given ina trapezoidABCD: lAB 1=IBC I = ICD I = a, I DA I = 2a. Takenrespectively on the straight lines ABand AD are points E and F, other than thevertices of the trapezoid, so that the pointof intersection of the altitudes of thetriangle CEF coincides with the point ofintersection of the diagonals of the trapezoidABCD. Find the area of the triangleCEF.* *18t. The altitude of a right triangle ABCdrawn to the hypotenuse AB is h, D beingits foot; M and N are the midpoints of theline segments AD and DB, respectively. 39. Sec. 1. Fundamental Facts 39Find the distance from the vertex C to thepoint of intersection of the altitudes of thetriangle CMN.182. Given an equilateral trapezoid withbases AD and Be: I AB I = I CD I = a,I AC I = IED I = b, IBe I = c, M anarbitrary point of the arc BC of the circlecircumscribed about ABeD. Find the ratio'IBMI+IMCIIAMI+IMDI183. Each lateral side of an isoscelestriangle is equal to 1, the base being equalto a. A circle is circumscribed about thetriangle. Find the chord intersecting thelateral sides of the triangle and dividedby the points of intersection into threeequal segments.184. MN is a diameter of a circle, IMNI=1, A and B are points on the circlesituated on one side from MN, C is a pointon the other semicircle. Given: A is themidpoint of semicircle, I MB I = 3/5, thelength of the line segment formed by theintersection of the diameter MN with thechords AC and Be is equal to a. What isthe greatest value of a?185. ABCD is a convex quadrilateral.M the midpoint of AB, N the midpoint ofCD. The areas of triangles ABN and CDMare known to be equal, and the area oftheir common part is 11k of the area ofeach of them. Find the ratio of the sidesBC and AD. 40. 40 Problems in Plane Geometry186. Given an equilateral trapezoid ABCD(AD II BC) whose acute angle at the largerbase is equal to 60, the diagonal beingequal to V3. The point M is found atdistances 1 and 3 from the vertices A andD, respectively. Find IMe I.187. The bisector of each angle of atriangle intersects the opposite side at apoint equidistant from the midpoints ofthe two other sides of the triangle. Does it,in fact, mean that the triangle is regular?188. Given in triangle are two sides:a and b (a > a b). Find the third side if itis known that a + ha ~ b + hb , where haand hb are the altitudes dropped on thesesides (ha the altitude drawn to the side a).189. Given a convex quadrilateral ABG'Dcircumscribed about a circle of diameter 1.Inside ABeD, there is a point M such thatI MA 12 + 1MB 12 + 1Me 12 + I MD 2 1=2. Find the area of ABCD.tOO. Given in a quadrilateral ABeD:I AB I = a, IBC I = b, ICD I = c, IDA 1=d; a2 + c2 =1= b2 + tP, c =1= d, M is apoint on BD equidistant from A and C.Find the ratio I BM 1 I MD I.191. The smaller side of the rectangleABeD is equal to 1. Consider four concentriccircles centred at A and passing, respectively,through B, C, D, and the intersectionpoint of the diagonals of the rectangleABeD. There also exists a rectangle with 41. Sec. 1. Fundamental Facts 4tvertices on the constructed circles ~onevertice per circle). Prove that there IS asquare whose vertices lie on the constructedcircles. Find its side.192. Given a triangle ABC. The perpendicularserected to AB and Be at their midpointsintersect the line AC at points Mand N such that IMN I = I AC I. Theperpendiculars erected to AB and AC attheir midpoints intersect BC at points Kand L such that 1KL I = ~ 1Be I. Findthe smallest angle of the triangle ABC.193. A point M is taken on the side ABof a triangle ABC such that the straightline joining the centre of the circle circumscribedabout the triangle ABC to themedian point of the triangle BCM is perpendicularto cu. Find the ratio IBM II BA I if I BC I I BA I = k.194. In an inscribed quadrilateral ABeDwhere IAB I = IBe I, K is the intersectionpoint of the diagonals. Find tAB Iif IBK I = b, I KD I = d.195. Give the geometrical interpretationsof equation (1) and systems (2), (3), and(4). Solve equation (1) and systems (2)and (3). In system (4) find x + y + z:(1)Vx2+a2-axV 3+-V y2+b2 - by Y3 . 42. 42 Problems in Plane Geometry+VX2+ y2_ XY V3=Va2+ b2 (a>O, b>O).(2) { x = VZ2_ a2 +Vy2_a2,y ==Vx2 - b2+Vz2- b2 ,Z = Vy2_ c2 +Vx2- c2.(3) x2+y2= (a-x2)+b2= d2+(b- y2.)(4) { x2+ xy + y2= a2,y2+yz+Z2 = b2,Z2+zx+x2 = a2 +b2196. The side of a square is equal to aand the products of the distances from theopposite vertices to a line l are equal toeach other. Find the distance from the centreof the square to the line l if it is known thatneither of the sides of the square is parallelto l.197. One of the sides in a triangle ABCis twice the length of the other and LB =2 LC. Find the angles of the triangle.198. A circle touches the sides AB andAC of an isosceles triangle ABC. Let Mbe the point of tangency with the side ABand N the point of intersection of the circleand the base BC. Find IAN I if IAM I =a, IBM I = b.199. Given a parallelogram ABCD inwhich I AB I = k I BC I, K and L arepoints on the line CD (K on the side CD), 43. Sec. t. Fundamental Facts 43and M is a point on BC, AD being thebisector of the angle KAL, AM the bisectorof the angle KAB, )BM I = a, IDL I =b. Find IAL I.200. Given a parallelogram ABCD. Astraight line passing through the vertex CIntersects the lines AB and AD at pointsl( and L, respectively. The areas of the,triangles KBC and CDL are equal to p~nd q, respectively. Find the area of theparallelogram ABeD.201. Given a circle of radius R and twopoints A and B on it such that IAB I = a.Two circles of radii x and y touch the given~$rcle at points A and B. Find: (a) thelength of the common external tangent tothe last circles if both of them touch the~,:en circle in the same way (either interil,lly or externally); (b) the length of the~JPmon internal tangent if the circle ofr~~ius x touches the given circle externally,while the circle of radius y touches thegiven circle internally..:. f202. Given in a triangle ABC: IAB I =f2~, I BC I = 13, I CA I = 15. Taken on~ side AC is a point M such that the !".'.'.ii.' of the circles inscribed in the triangles . AI and ReM are equal. Find the ratio,!1M I : IMe ).i.",. 203. The radii of the circles inscribed and circumscribed about a triangle areiiual to rand R, respectively. Find the.'tea of the triangle if the circle passing 44. 44 Problems in Plane Geometrythrough the centres of the inscribed andcircumscribed circles and the intersectionpoint of the altitudes of the triangle isknown to pass at least through one of thevertices of the triangle.204. Given a rectangle AI1CD whereI AB I = 2a, I BC I = a V2. On the sideAR, as on diameter, a semicircle is constructedexternally. Let M be an arbitrarypoint on the semicircle, the line MD intersectAB at N, and the line Me at L. Find1AL 12 + I BN 12 (Fermat's" problem).205. Circles of radii Rand r touch eachother internally. Find the side of theregular triangle, one vertex of which coincideswith the point of tangency, and theother two, lying on the given circles.206. Two circles of radii Rand r (R > r)touch each other externally at a point A.Through a point B taken on the largercircle a straight line is drawn touching thesmaller circle at C. Find I Be I if IAB I =a.207. In a parallelogram ABeD there arethree pairwise tangent circles" *; one 0 fthem also touches the sides AB and Be, thesecond the sides AB and AD, and the thirdthe sides Be and AD. Find the radius ofthe third circle if the distance between the Fermat, Pierre de (t60t-1665), a Frenchamateur mathematician. Any two of them have a point of tangency. 45. Sec. 1. Fundamental Facts 45points of tangency on the side AB is equalto a.208. The diagonals of the quadrilateralABeD intersect at a point M, the anglebetween them equalling Ct. Let 01' 02' 03'0 denote the centres of the circles circumscribedabout the triangles ARM, HeM,CDM, DAM, respectively. Determine theratio of the areas of the quadrilateralsABeD and 123.. 209. In a parallelogram whose area is S,the bisectors of its interior angles are drawnto intersect one another. The area of thequadrilateral thus obtained is equal to Q.Find the ratio of the sides of the parallelogram.2tO. In a triangle ABC, a point M istaken on the side AC and a point N on theade BC. The line segments AN and BMiBtersect at a point O. Find the area of the.~angle CMN if the areas of the triangles{)MA, DAB, and OBM are equal to 81, S2'ahd S3' respectively.211. The median point of a right trianglelies on the circle inscribed in this triangle.B'i;nd the acute angles of the triangle.212. The circle inscribed in a trianglekBC divides the median BM into threeequal parts. Find the ratio I BC I I CA I-lAB I-213. In a triangle ABC, the midperpendic-alar to the side AB intersects the line ACat M, and the midperpendicular to the side 46. 46 Problems in Plane GeometryAC intersects the line AB at N. It is knownthat I MN I = I BC I and the line MNis perpendicular to the line BC. Determinethe angles of the triangle ABC.214. The area of a trapezoid with basesAD and BC is S, IAD I : I BC I = 3;situated on the straight line intersecting theextension of the base AD beyond the pointD there is a line segment EF such thatAE II DF, BE II CF and I AE I : IDF I =I CF I IBE I = 2. Determine the area ofthe triangle EFD.215. In a triangle ABC the side BC isequal to a, and the radius of the inscribedcircle is r. Find the area of the triangle ifthe inscribed circle touches the circle constructedon BC as diameter.216. Given an equilateral triangle ABCwith side a, BD being its altitude. A secondequilateral triangle BDCl is constructedon BD, and a third equilateral triangleBDICt is constructed on the altitude BD!of this triangle. Find the radius of the circlecircumscribed about the triangle CClC2 Prove that its centre is found on one of thesides of the triangle ABC (C2 is situatedoutside the triangle ABC).217. The sides of a parallelogram areequal to a and b (a =1= b). Straight lines aredrawn through the vertices of the obtuseangles of this parallelogram perpendicularto its sides. When intersecting, these linesform a parallelog~am similar to the given 47. Sec. t. Fundamental Facts 47one. Find the cosine of the acute angle ofthe given parallelogram.218. Two angle bisectors KN and LPintersecting at a point Q are drawn in atriangle KLM. The line segment PN hasa length of 1, and the vertex M lies on theeircle passing through the points N, P,and Q. Find the sides and angles of thetriangle PNQ.. 219. The centre of a circle of radius rtouching the .sides AB, AD, and Be islocated on the diagonal AC of a convexfluadrilateral ABeD. The centre of a circleof' the same radius r touching the sidesBe, CD, and AD is found on the diagonalBD. Find the area of the quadrilateral~BCD if the indicated circles touch eachother externally.220. The radius of the circle circumscribed.bout an acute-angled triangle ABC is equallo t. The centre of the circle passing throughthe vertices A, C, and the intersection point"of the altitudes of the triangle ABC is known:~ lie on this circle. Find I AC I.t i 221. Given a triangle ABC in which"ints M, N, and P are taken: M and N~ the sides AC and BC, respectively, P;.,. the line segment MN such that IAM Ij~.MC I = I CN I : INB I = IMP I : I PN Ilind the area of the triangle ABC if the.-eas of the triangles AMP and BNP are T~d. Q, respectively.~'~ 222. Given a circle of radius R and a 48. 48 Problems in Plane Geometrypoint A at a distance a from its centre(a > R). Let K denote the point of thecircle nearest to the point A. A secant linepassing through A intersects the circle atpoints M and N. Find IMN I if the areaof the triangle KMN is S.223. In an isosceles triangle ABC(I AB I = ,BC I), a perpendicular to AEis drawn through the end point E of theangle bisector AE to intersect the extensionof the side AC at a point F (C lies bet-veenA and F). It is known that I AC I = 2m,I FC I = m/4. Find the area of the triangleABC.224. Two congruent regular trianglesABC and CDE with side 1 are arrangedon a plane so that they have only one commonpoint C, and the angle BCD is less thann/3. K denotes the midpoint of the sideAC, L the midpoint of GE, and M the midpointof RD. The area of the triangle KLMis equal to V3/5. Find IBD I.225. From a point K situated outside acircle with centre 0, two tangents KMand KN (M and N points of tangency) aredrawn. A point C (I Me I < I CN I) istaken on the chord MN. Drawn through thepoint C perpendicular to the line segmentOC is a straight line intersecting the linesegment N K at B. The radius of the circleis known to be equal to R, LMKN = a,1 MC I = b. Find I CB I 49. Sec. t. Fundamental Facts 49226. A pentagon ABCDE is inscribed ina circle. The points M, Q, N, and P are thefeet of the perpendiculars dropped from thevertex E of the sides AB, Be, CD (or theirextensions), and the diagonal AD, respectively.It is known that I EP I = d, and theratio of the areas of the triangles MQE andPNE is k. Find IEM f.227. Given a right trapezoid. A straightline, parallel to the bases of the trapezoidseparates the latter into two trapezoidssuch that a circle can be inscribed in eachof them. Determine the bases of the originaltrapezoid if its lateral sides are equal to cand d (d > c).228. Points P and Q are chosen on thelateral sides KL and M N of an equilateraltrapezoid KLMN, respectively, such thatthe line segment PQ is parallel to the basesof the trapezoid. A circle can be inscribedin each of the trapezoids KPQN andPLMQ, the radii of these circles beingequal to Rand r, respectively. Determinethe bases I LM I and I KN I229. In a triangle ABC, the bisector ofthe angle A intersects the side BC at apoint D. It is known that lAB I-IBD 1=a, I AC I + I CD I = b. Find I AD I230. Using the result of the precedingproblem, prove that the square of thebisector -of the triangle is equal to the productof the sides enclosing this bisectorminus the product of the line segments of4-01557 50. 50 Problems in Plane Geometrythe third side into which the latter is dividedby the bisector.231. Given a circle of diameter A B. A secondcircle centred at A intersects the firstcircle at points C and D and its diameter atE. A point M distinct from the points Cand E is taken on the arc CE that does notinclude the point D. The ray BM intersectsthe first circle at a point N. It is known thatI CN I = a, IDN I = b. Find I MN I232. In a triangle ABC, the angle B is'Jt/4, the angle C is :1/6. Constructed on themedians BN and CN as diameters are circlesintersecting each other at points Pand Q. The chord PQ intersects the sideBC at a point D. Find the ratio I BD IIDC I.233. Let AB denote the diameter of acircle, 0 its centre, AB = 2R, C a pointon the circle, M a point on the chord AC.From the point M, a perpendicular MNis dropped on AB and another one is erectedto AC intersecting the circle at L (theline segment CL intersects AB). Find thedistance between the midpoints of AOand CL if I AN I = a.234. A circle is circumscribed about atriangle ABC. A tangent to the circlepassing through the point B intersects theline AC at M. Find the ratio IAM IIMe I if IAB I : I BC I = k.235. Points A, B, C, and D are situatedin consecutive order on a straight line, 51. Sec. f. Fundamental Facts 5twhere I AC I = a, IAB I, IAD I = PIAB IAn arbitrary circle is described through Aand B, CM and DN being two tangents tothis circle (M and N are points on the circlelying on opposite sides of the line AB).In what ratio is the line segment AB dividedby the line MN?236. In a circumscribed quadrilateralABeD, each line segment from A to thepoints of tangency is equal to a, and eachline segment from C to the points of tangencyis b. What is the ratio in which thediagonal AC is divided by the diagonalBD?237. A point K lies on the base AD of thetrapezoid ABeD. such that IAK I =A IAD I. Find the ratio IAM I : I MD I,where M is the point of intersection of thebase AD and the line passing through theintersection points of the lines AB and CDand the lines BK and AC.Setting A = tin (n = 1, 2, 3, .], di-videa given line segment into n equal partsusing a straight edge only given a straightline parallel to this segment.238. In a right triangle ABC with thehypotenuse AB equal to c, a circle is constructedon the altitude CD as diameter.Two tangents to this circle passing throughthe points A and B touch the circle at pointsM and N, respectively, and, when extended,intersect at a point K. Find I MK I.239. Taken on the sides AB, Be and CA4* 52. 52 Problems in Plane Geometryof a triangle ABC are points Cl , Al and BIsuch that I ACt I : I CIB I = I BAt I :I Al CI = I CBt I: I BtA I = k. Taken onthe sides AtBI , BICI , and CIA I are pointsC2 , A 2 , and B 2 , such that I A IC2 II GtB I , = I BIA 2 I I AtCI I = I c.s, II B 2A I I = 11k. Prove that the triangleAtB2C2 is similar to the triangle ABC andfind the ratio of similitude.240. Given in a triangle ABC are theradii of the circumscribed (R) and inscribed(r) circles. Let At, Bt , Cl denote the pointsof intersection of the angle bisectors of thetriangle ABC and the circumscribed circle.Find the ratio of the areas of the trianglesABC and AIBICt .241. There are two triangles with correspondinglyparallel sides and areas 81 and 8 2 ,one of them being inscribed in a triangleABC, the other circumscribed about thist-riangle. Find the area of the triangle ABC.242. Determine the angle A of the triangleABC if the bisector of this angle is perpendicularto the straight line passingthrough the intersection point of the alti..tudes of this triangle and the centre of thecircumscribed circle.243. Find the angles of a triangle if thedistance between the centre of the circumscribedcircle and the intersection pointof the altitudes is one-half the length of thelargest side and equals the smallest side.244. Given a triangle ABC. A point D 53. Sec. 1. Fundamental Facts 53is taken on the ray BA such that I BD I =I BA I + IAC I Let K and M denote twopoints on the rays BA and Be, respectively,such that the area of the triangle BDMis equal to the area of the triangle BCK.Find LBKM if LBAC == Ct.245. In a trapezoid ABeD, the lateralside AB is perpendicular to AD and BC,and I AB I = VI AD 11 BC I. LetE denotethe point of intersection of the nonparallelsides of the trapezoid, 0 the intersectionpoint of the diagonals and M the midpointof AB. Find LEOM.246. Two points A and B and two straightlines intersecting at 0 are given in a plane.Let us denote the feet of the perpendicularsdropped from the point A on the given linesby M and N, and the feet of the perpendicularsdropped from B by K and L, respectively.Find the angle between the lines MNand KL if LAOB = Ct ~ 90.247. Two circles touch each other internallyat a point A. A radius OB touchingthe smaller circle at C is drawn from thecentre 0 of the larger circle. Find the angleBAC.248. Taken inside a square ABCD is apoint Msuch thatLMAB = 60, LMCD =15. Find LMBC.249. Given in a triangle ABC are twoangles: LA = 45 and LB = 15 Takenon the extension of the side AC beyond the 54. 54 Problems in Plane Geometrypoint C is a point M such that I CM I =2 I AC I. Find LAMB.250. In a triangle ABC, LB = 60 andthe bisector of the angle A intersects BCat M. A point K is taken on the side ACsuch that LAMK = 30. Find LOKe,where 0 is the centre of the circle circumscribedabout the triangle AMC.251. Given a triangle ABC in which1AB I = IAC I, LA = 80. (a) A pointM is taken inside the triangle such thatLMBC = 30, LMCB = 10. Find LAMC.(b) A point P is taken outside the trianglesuch that LPBC = LPCA = 30, andthe line segment BP intersects the sideAC. Find LPAC.252. In a triangle ABC, LB = 100,LC = 65; a point M is taken on AB suchthat LMCB = 55, and a point N is takenon AC such that LNBC = BO. FindLNMC.253. In a triangle ABC, I AB I = I BC I,LB = 20. A point M is taken on the sideAB such that LMCA = 60, and a point Non the side CB such that LNA C = 50.Find LNMC.254. In a triangle ABC, LB = 70,LC = 50. A point M is taken on the sideAB such that LMCB = 40, and a point Non the side AC such that LNBC = 50.Find LNMC.255. Let M and N denote the points oftangency of the inscribed circle with the 55. Sec. 1. Fundamental Facts 55sides Be and BA of a triangle ABC, K theintersection point of the bisector of theangle A and the line MN. Prove thatLAKC = 90.256. Let P and Q be points of the circlecircumscribed about a triangle ABC suchthat I PA 12 = I PB 11 PC I, IQA 12 =I QR I IQC I (one of the points is on thearc AB, the other on the arc A C). FindLPAB - LQAC if the difference betweenthe angles Band C of the triangle ABCis (7,.257. Two fixed points A and B are takenon a given circle and ........AB = (7,. An arbitrarycircle passes through the points A andB. An arbitrary line 1 is also drawn throughthe point A and intersects the circles atpoints C and D different from B (the pointC is on the given circle). The tangents tothe circles at the points C and D (C and Dthe points of tangency) intersect at M;N is a point on the line 1such that I eN I =I AD I, IDN I = I CA I What are thevalues the LCMN can assume?258. Prove that if one angle of a triangleis equal to 120, then the triangle formedby the feet of its angle bisectors is rightangled.259. Given in a quadrilateral ABCD:LDAB = 150, LDAC + LARD = 120,LDBC - LARD = 60. Find LBDe. 56. 56 Problems in Plane Geometry * *260. Given in a triangle ABC: I AB I =1. I AC I = 2. Find I BC I if the bisectorsof the exterior angles A and C are knownto be congruent (i.e., the line segment ofthe bisectors from the vertex to the intersectionpoint with the straight line includingthe side of the triangle opposite to theangle).261. A point D is taken on the side CBof a triangle ABC such that I CD I =ex lAC [. The radius of the circle circumscribedabout the triangle ABC is R. Findthe distance between the centres of thecircles circumscribed about the trianglesABC and ADB.262. A circle is circumscribed about 8right triangle ABC (LC = 90). Let CDdenote the altitude of the triangle. A circlecentred at D passes through the midpoint ofthe arc AB and intersects AB at M. FindJ cu I if I AB I = c.263. Find the perimeter of the triangleABC if IBC I = a and the segment of thestraight line tangent to the inscribed circleand parallel to Be which is enclosed insidethe triangle is b.264. Three straight lines parallel to thesides of a triangle and tangent to the inscribedcircle are drawn. These lines cutoff three triangles from the given one. Theradii of the circles circumscribed about them 57. Sec. 1. Fundamental Facts 57are equal to R1, R t , and R a Find the radiusof the circle circumscribed about the giventriangle.265. Chords AB and AC are drawn in acircle of radius R. A point M is taken onAB or on its extension beyond the point B,the distance from M to the line AC beingequal to I AC I. Analogously a point Nis taken on AC or on its extension beyondthe point C, the distance from N to) theline AB being equal to I AB I. Find MN.266. Given a circle of radius R centredat O. Two other circles touch the givencircle internally and intersect at points Aand B. Find the sum of the radii of thesetwo circles if LOAB = 90.267. Two mutually perpendicular intersectingchords are drawn in a circle of radiusR. Find (a) the sum of the squares ofthe four segments of these chords intowhich they are divided by the point of intersection;(b) the sum of the squares of thechords if the distance from the centre ofthe circle to the point of their intersectionis equal to a.268. Given two concentric circles of radiirand R (r < R). A straight line is drawnthrough a point P on the smaller circle tointersect the larger circle at points BandC. The perpendicular to Be at the point Pintersects the smaller circle at A. FindI PA 1 2 + I PB 1 2 + I PC 21269. In a semicircle, two intersecting 58. 58 Problems in Plane Geometrychords are drawn from the end points of thediameter. Prove that the sum of the productsof each chord segment that adjoinsthe diameter by the entire chord is equalto the square of the diameter.270. Let a, b, C and d be the sides of aninscribed quadrilateral (a be opposite to c),ha , hb , he' and hd the distances from the centreof the circumscribed circle to' the correspondingsides. Prove that if the centre ofthe circle is inside the quadrilateral, thenahc + cha = bhd + dh b 271. Two opposite sides of a quadrilateralinscribed in a circle intersect at pointsP and Q. Find I PQ I if the tangents to thecircle drawn from P and Q are equal to aand b, respectively.272. A quadrilateral is inscribed in a circleof radius R. Let P, Q, and M denotethe points of intersection of the diagonals'of this quadrilateral with the extensions ofthe opposite sides, respectively. Find thesides of the triangle PQM if the distancesfrom P, Q, and M to the centre of the circleare a, b, and c, respectively.273. A quadrilateral ABeD is circumscribedabout a circle of radius r. The point oftangency of the circle with the side ABdivides the latter into segments a and b,and the point at which the circle touchesthe side AD divides that side into segmentsa and c. What are the limits of change of r?274. A circle of radius r touches inter- 59. Sec. 1. Fundamental Facts 59nally a circle of radius R, A being the pointof tangency. A straight line perpendicularto the centre line intersects one of the circlesat B, the other at C. Find the radiusof the circle circumscribed about the triangleABC.275. Two circles of radii Rand r intersecteach other, A being one of the points of intersection,BC a common tangent (B andC points of tangency). Find the radius ofthe circle circumscribed about the triangleABC.276. Given in a quadrilateral ABCD:I AB I = a, I AD I = b; the sides BC, CD,and AD touch a circle whose centre is in themiddle of AB. Find I BC I .277. Given in, an inscribed quadrilateralABCD: IAB I = a, I AD I = b (a >b). Find I BC I if Be, CD, and ADtouch a circle whose centre lies on A B.* *278. In a convex quadrilateral ABCD,IAB I = IAD I. Inside the triangleABC, a point M is taken such thatLMBA = LADe, LMCA = LACD.Find L MAC if L BAG = cx, L ADG LACD = q>, IAM I < I AB I279. Two intersecting circles are inscribedin the same angle, A being the vertex ofthe angle, B one of the intersection pointsof the circles, C the midpoint of the chordwhose end points are the points of tangency 60. 60 Problems in Plane Geometryof the first circle with the sides of the angle.Find the angle ABC if the common chordcan be observed from the centre of thesecond circle at an angle a.280. In an isosceles triangle ABC,IAC I = 1Be I, BD is an angle bisector,BDEF is a rectangle. Find L BAFif L BAE = 1200281. A circle centred at 0 is circumscribedabout a triangle ABC. A tangent to the circleat point C intersects the line bisectingthe angle B at a point K, the angle BKC beingone-half the difference between the tripleangle A and the angle C of the triangle.The sum of the sides AC and AB is equalto 2 + V3" and the sum of the distancesfrom the point 0 to the sides AC and ABequals 2. Find the radius of the circle.282. The points symmetric to the verticesof a triangle with respect to the oppositesides represent the vertices of the trian-glewith sides VB, VB, V14. Determine thesides of the original triangle if their lengthsare different.283. In a triangle ABC, the angle betweenthe median and altitude emanating fromthe angle A is a, and the angle between themedian and altitude emanating from Bis ~. Find the angle between the median andaltitude emanating from the angle C.284. The radius of the circle circumscribedabout a triangle is R. The distance 61. Sec. 1. Fuudamental Facts 61from the centre of the circle to the medianpoint of the triangle is d. Find the productof the area of the given triangle and thetriangle formed by the lines passingthrough its vertices perpendicular to themedians emanating from those vertices.285. The points AI' A3 and As are situatedon one straight line, and the points A 2'A., and A 8 on the other intersecting thefirst line. Find the angles between theselines if it is known that the sides of thehexagon A IA 2A sA.Af)A. (possibly, a selfintersectingone) are equal to one another.286. Two circles with centres 0 1 and O2touch internally a circle of radius R centredat O. It is known that I 0102 I = a.A straight line touching the first two circlesand intersecting the line segment 0 102intersects their common external tangentsat points M and N and the larger circle atpoints A and B. Find the ratio I AB II MNI if (8) the line segment 0102 containsthe point 0; (b) the circles with centres0 1 and O2 touch each other",287. The circle inscribed in a triangleABC touches the side AC at a point M andthe side BC at N; the bisector of the angleA intersects the line MN at K, and the bisectorof the angle B intersects the line MNat L. Prove that the line segments MK,N L, and KL can form a triangle. Find thearea of this triangle if the area of thetriangle ABC is S, and the angle C is a. 62. 62 Problems in Plane Geometry288. Taken on the sides AB and BC of asquare are two points'M and N such thatIBM I + IBN I = I AB I. Prove thatthe lines DM and DN divide the diagonalAC into three line segments which can forma triangle, one angle of this triangle beingequal to 60.289. Given an isosceles triangle ABC,I AB I = I BC I , AD being an angle bisector.'(he perpendicular erected to ADat D intersects the extension of the sideAC at a point E; the feet of the perpendicularsdropped from Band D on AC are pointsM and N, respectively. Find I MN I ifIAE I = a.290. Two rays emanate from a point Aat an angle Cl. Two points Band BI aretaken on one ray and two points C and CIon the other. Find the common chord of thecircles circumscribed about the trianglesABC and ABICI if I AB I - I AC I =I ARI I - I AC 1 I = a.291. Let 0 be the centre of a circle, Ca point on this circle, M the midpoint ofOC. Points A and B lie on the circle on thesame side of the line OC so that L AMO =L BMC. Find I AB I if IAM I -IBM I = a.292. Let A, B, and C be three points lyingon the same line. Constructed on AR,BC, and AC as diameters are three semicircleslocated on the same side of the line.The centre of a circle touching the three 63. Sec. f. Fundamental Facts 63semicircles is found at a distance d from theline AC. Find the radius of this circle.293. A chord AB is drawn in a circle ofradius R. Let M denote an arbitrary pointof the circle. A line segment MN ( IMN I =R) is laid off on the ray M A and on theray MB a line segment M K equal to thedistance from M to the intersection pointof the altitudes of the triangle MAB. FindINK I if the smaller of the arcs subtendedby AB is equal to 2a.294. The altitude dropped from the rightangle of a triangle on the hypotenuse separatesthe triangle into two triangles in eachof which a circle is inscribed. Determine theangles and the area of the triangle formed bythe legs of the original triangle and the linepassing through the centres of the circlesif the altitude of the original triangle is h.295. The altitude of a fight triangledrawn to the hypotenuse is equal to h.Prove that the vertices of the acute anglesof the triangle and the projections of thefoot of the altitude on the legs all lie on thesame circle. Determine the length of thechord cut by this circle on the line containingthe. altitude and the segments of thechord into which it is divided by the hypotenuse.296. A circle of radius R touches aline l at a point .A, AB is a diameterof this circle, Be is an arbitrary chord.Let D denote the foot of the perpendicular 64. Problems in Plane Geometrydropped from C on AB. A point E lies on theextension of CD beyond the point D, andI ED I = I BC I The tangents to the cir-cle,passing through E, intersect the linel at points K and N. Find I KN I297. Given in a convex quadrilateralABeD: IAB I = a, I AD I = b, I Be I ~p - a, IDC I = p - b. Let 0 be thepoint of intersection of the diagonals. Letus denote the angle BA C by a. What doesI AO I tend to as a ~ O? 65. Section 2Selected Problems andTheorems of Plane GeometryCarnot's Theoremt. Given points A and B. Prove that thelocus of points M such that IAM I 2 IMB I 2 = k (where k is a given number)is a straight line perpendicular to AB.2. Let the distances from a point M tothe vertices A, B, and C of a triangle ABCbe a, b, and c, respectively. Prove that thereis no d =1= 0 and no point on the planefor which the distances to the vertices in thesame order can be expressed by the numbersVa2 + d, Vb2 + d, Vc2 + d.3. Prove that for the perpendiculars drop-pedfrom the points At, B1 , and Cion thesides Be, CA, and AB of a triangle ABCto intersect at the same point, it is necessaryand sufficient thatI AlB 12- I BCI 12 + I CIA 122+_ I ABI 11BtC 12- I CAl 1 2 = 0 (Carnot's theorem).4. Prove that if the perpendiculars drop...ped from the points AI' Bt , and Ct on thesides BC, CA, and AB of the triangle ABC,respectively, intersect at the same point,then the perpendiculars dropped from the5 -01557 66. 66 Problems in Plane Geometrypoints A, B, and C on the lines BICI, CIA I,and AIBI also intersect at one point.5. Given a quadrilateral ABeD. Let AI,Bl , and C1 denote the intersection pointsof the altitudes of the triangles BCD,ACD, and ABD. Prove that the perpendicularsdropped from A, B, and C on thelines BICI , CIAI , and AtBI , respectively,intersect at the same point.6. Given points A and B. -Prove that thelocus of points M such that k I AM 1 2 +l I BM I 2 = d (k, l, d given numbers,k + l =1= 0) is either a circle with centre onthe line AB or a point or. an empty set.7. Let At, A 2 , , An be fixed pointsand kl , k 2 , . ., k n be given numbers. Thenthe locus of points M such that the sumkl I AIM 1 2 + k 2 I A 2M 1 2 + +k n I A nM I 2 is constant is: (a) a circle, apoint, or an empty set if kl + k2 +. +k n =1= 0; (b) a straight line, an empty set,or the entire plane if kl + k 2 + +k; -= o.8. Given a circle and a point A outsidethe circle. Let a circle passing through Atouch the given circle at an arbitrary pointB, and the tangents to the second circlewhich are drawn through the points A andB intersect at a point M. Find the locus ofpoints M.9. Given points A and .B. Find the locusof points M such that IAM I I MB I =k =1= 1. 67. Sec. 2. Selected Problems 6710. Points A, B, and C lie on a straightline (B between A and C). Let us take an arbitrarycircle centred at B and denote by Mthe intersection point of the tangents drawnfrom A and C to that circle. Find the locusof points M such that the points of tangencyof straight lines AM and CM with the circlebelong to the open intervalsAMand GM.1I , Given two circles. Find the locus ofpoints M such that the ratio of the lengthsof the tangents drawn from M to the givencircles is a constant k.12. Let a straight line intersect one circleat points A and B and the other atpoints C and D. Prove that the intersectionpoints of the tangents to the first circlewhich are drawn at points A and B and thetangents drawn to the second circle atpoints C and D (under consideration are theintersection points of tangents to distinctcircles) lie on a circle whose centre is foundon the straight line passing through the centresof the given circles.13. Let us take three circles each of whichtouch one side of a triangle and the extensionsof two other sides. Prove that the perpendicularserected to the sides of the triangleat the points of tangency of these circlesintersect at the same point.14. Given a triangle ABC. Consider allpossible pairs of points M1 and M2 such thatIAMt I : IBMt I : I CMt I = IAMI I :IBM2 I I CM2 I Prove that the lines5* 68. Problems in Plane GeometryMIMI pass through the same fixed pointin the plane.t5. The distances from a point M to thevertices A, B, and C of a triangle are equalto 1, 2, and 3, respectively, and from apoint M1 to the same vertices to 3,V15, 5,respectively. Prove that the straight lineMMl passes through the centre of the circlecircumscribed about the triangle ABC.16. Let AI' Bl , Gl denote the feet of theperpendiculars dropped from the verticesA, B, and C of a triangle ABC on the line l,Prove that the perpendiculars dropped fromAI' Bl , and Cion Be, CA, and AB, respectively,intersect at the same point.17. Given a quadrilateral triangle ABCand an arbitrary point D. Let AI' Bl , andC1 denote the centres of the circles inscribedin the triangles BCD, CAD, and ABD,respectively. Prove that the perpendicularsdropped from the vertices A, B, and Con BlGl, CIAt , and AIBl , respectively, intersectat the same point.18. Given three pairwise intersecting circles.Prove that the three common chords ofthese circles pass through the same- point.19. Points M and N are taken on lines ABand AC, respectively. Prove that the commonchord of two circles with diameterseM and BN passes through the intersectionpoint of the altitudes of the triangle ABC.20. A circle and a point N are given in a 69. Sec. 2. Selected Problems 69plane. Let AB be an arbitrary chord of thecircle. Let M denote the point of intersectionof the line AB and the tangent at thepoint N to the circle circumscribed aboutthe triangle ABN. Find the locus of pointsM.21. A point A is taken inside a circle.Find the locus of the points of intersectionof the tangents to the circle at the end pointsof all possible chords passing through thepoint A.22. Given numbers a, p, y, and k. Letx, y, I denote the distances from a point Mtaken inside a triangle to its sides. Provethat the locus of points M such that ax +py + "1 = k is either an empty set ora line segment or coincides with the set ofall points of the triangle.23. Find the locus of points M situatedinside a given triangle and such that thedistances from M to the sides of the giventriangle can serve as sides of a certain triangle.24. Let AI' BI , and C1 be- the midpointsof the sides BC, CA, and AB of a triangleABC, respectively. Points A 2 , B2 , and C2are taken on the perpendiculars droppedfrom a point M on the sides BC, CA, andAR, respectively. Prove that the perpendicularsdropped from At, Bt , and Cionthe lines BtC2 , C2A'}" and A tB2 , respectively,intersect at the same point.25. Given a straight line l and three 70. 70 Problems in Plane Geometrylines It, l2' and i, perpendicular to l. LetA, B, and C denote three fixed points onthe line l, At an arbitrary point on ll' B1an arbitrary point on ll' C1 an arbitrarypoint on ls. Prove that if at a certain arrangementof the points At, BI , and C1 the perpendicularsdropped from A, B, and Con the lines BIGI , CIAI , and AtBI , respectively,intersect at one and the same point,then these perpendiculars meet in the samepoint at any arrangement of At, Bt , G126. Let AAI , BBI , GCI be the altitudes ofa triangle ABC, AI' Bi , and C2 be the projectionsof A, B, and C on BIGI, CIAI,and AIBI , respectively. Prove that the perpendicularsdropped from A 2 , B2 , and C"on BC, CA~, and AR, respectively, intersectat the same point.Ceva's* and Menelaus'** Theorems.Affine Problems2:1. Prove that the area of a triangle whosesides are equal to the medians of a giventriangle amounts.. to 3/4 of the area of thelatter. Ceva, Giovanni (t647-t 734). An Italianmathematician who gave static and geometricproofs for concurrency of straight lines throughvertices of triangles. Menelaus of Alexandria (first cent. A.D.).A geometer who wrote several books on plane andspherical triangles, and circles. 71. Sec. 2. Selected Problems 7128. Given a parallelogram ABCD. Astraight line parallel to BC intersects ABand CD at points E and F, respectively,and a straight line parallel to AB intersectsBC and DA at points G and H, respectively.Prove that the lines EH, GF, and BD eitherintersect at the same point or are parallel.29. Given four fixed points on a straightline l : A, B, C, and D. Two parallel linesare drawn arbitrarily through the pointsA and B, another two through C and D.The lines thus drawn form a parallelogram.Prove that the diagonals of that parallelogramintersect l at two fixed points.30. Given a quadrilateral ABeD. Leto be the point of intersection of the diagonalsAC and BD, M a point on AC such thatIAM ( = IOC (, N a point on BD suchthat I BN I = I OD I , K and L the midpointsof AC and BD. Prove that the linesML, N K, and the line joining the medianpoints of the triangles ABC and ACD intersectat the same point.31. Taken on the side Be of a triangleABC are points Al and A 2 which are symmetricwith respect to the midpoint of BC.In similar fashion taken on the side AC arepoints BI and B2 , and on the side ABpoints C1 and C2 Prove that the trianglesAIBICl and A 2B2C2 are equivalent, andthe centres of gravity of the trianglesAlBIC!, A 2B2C2 , and ABC are collinear.32. Drawn through the intersection point 72. 72 Problems in Plane GeometryM of medians of a triangle ABC is astraight line intersecting the sides AB andAC at points K and. L, respectively, andthe extension of the side BC at a point P (Clying between P and B). Prove that I~KI =1 1IMLI + IMPI .33. Drawn through the intersection pointof the diagonals of a quadrilateral ABCDis a straight line intersecting AB at a pointM and CD at a point N. Drawn throughthe points M and N are lines parallel toCD and AR, respectively, intersecting ACand BD at points E and F. Prove that BEis parallel to CF.34. Given a quadrilateral ABCD. Takenon the lines AC and BD are points K andM, respectively, such that BK II AD andAM II BC. Prove that KM II CD.35. Let E be an arbitrary point taken onthe side AC of a triangle ABC. Drawnthrough the vertex B of the triangle is anarbitrary line l. The line passing through thepoint E parallel to Be intersects the linel at a point N, and the line parallel to ABat a point M. Prove that AN is parallel toCM. 36. Each of the sides of a convex quadrilateralis divided into (2n + 1) equal parts.The division points on the opposite sides 73. Sec. 2. Selected Problems 73are joined correspondingly. Prove that thearea of the central quadrilateral amounts to1/(2n + 1)2 of the area of the entire quadrilateral.37. A straight line passing through themidpoints of the diagonals AC and BC of aquadrilateral ABeD intersects its sides ABand DC at points M an N, respectively.Prove that 8 DCM = SASH.38. In a parallelogram ABeD, the verticesA, B, C, and D are joined to the midpointsof the sides CD, AD, AB, andBe, respectively. Prove that the area ofthe quadrilateral formed by these line segmentsis 1/5 of the area of the parallelogram.39. Prove that the area of the octagonformed by the lines joining the vertices of aparallelogram to the midpoints of the oppositesides is 1/6 of the area of the parallelogram.40. Two parallelograms ACDE andBCFG are constructed externally" on thesides AC and BC of a triangle ABC. Whenextended, DE and FD intersect at a pointH. Constructed on the side AB is a parallelogramABML, whose sides AL and BM areequal and parallel to HC. Prove that theparallelogram ARML is equivalent to thesum of the parallelograms constructed onAC and Be. Here and elsewhere, such a notation symbolizesthe area of the figure denoted by the subscript. 74. 74 Problems in Plane Geometry41. Two parallel lines intersecting the largerbase are drawn through the end pointsof the smaller base of a trapezoid. Thoselines and the diagonals of the trapezoid separatethe trapezoid into seven triangles andone pentagon. Prove that the sum of theareas of the triangles adjoining the lateralsides and the smaller base of the trapezoidis equal to the area of the pentagon.42. In a parallelogram ABCD, a point Elies on the line AB, a point F on the lineAD (B on the line segment AE, D on AF),K being the point of intersection of thelines ED and FB. Prove that the quadrilateralsABKD and CEKF are equivalent.* * *43. Consider an arbitrary triangle ABC.Let AI' B1, and C1 be three points on thelines BC, CA, and AB, respectively. Usingthe following notationR = I ACt I'. I BA I I . I CB1 II CtB I I AIC I I BlA I 'R* _ sin LACCl sin LBAAI sin LCBB1- sin LelCB sin LA1AC sin LBIBA 'prove that R = R*.44. For the lines AAl t BBt , eel to meetin the same point {or for all the three to beparallel), it is necessary and sufficient thatR = 1 (see the preceding problem), and ofthree points At., B., Ct the one or all the 75. Sec. 2. Selected Problems 75three lie on the sides of the triangle ABC,and not on their extensions (Ceva's theorem).45. For the points AI' BI , C1 to lie on thesame straight line, it is necessary and sufficientthat R = 1 (see Problem 43, Sec. 2),and of three points AI' Bt t CI no points ortwo lie on the sides of the triangle ABC,and not on their extensions (Menelaus'theorem).Remark. Instead of the ratio : ~~~: andthe other two, it is possible to considerthe ratios of directed line segments whichAC are denoted by Cl~ and defined as foI-lows. ICACtBt I.:.= II ACCtBt II ' CACtBt I-S POSIitrive---. ~when the vectors ACt and CiB are in thesame direction and ~~~ negative if thesevectors are in opposite directions.(~~~ has sense only for points situatedon the same straight line.) It is easilyseen that the ratio ~~; is positive if thepoint C1 lies on the line segment AB andthe ratio is negative if C, is outside AB.Accordingly, instead of R, we shall considerthe product of the ratios of directed linesegments which is denoted by If. Further,we introduce the notion of directed angles. 76. 76 Problems in Plane GeometryFor instance, by 4ACCI we shall understandthe angle through which we have to rotateCA about C anticlockwise to bring the rayCA into coincidence with the ray eel.Now, instead of R* we shall consider theproduct of the ratios of the sines of directedangles j .Now, we have to reformulate Problems43, 44, and 45 of this Section- in the followingway:43*. Prove that j = j .44*. For the lines AAl t BBt , eel to meetin the same point (or to be parallel), it isnecessary and sufficient that R= 1(Ceva's theorem). 45*. For the points AI' Bit C1 to be collinear,it is necessary and sufficient thatR= - 1 (Menelaus' theorem).46. Prove that if three straight lines, passingthrough the vertices of a triangle, meetin the same point, then the lines symmetricto them with respect to the correspondingangle bisectors of the triangle alsointersect at one point or are parallel.47. Let 0 denote an arbitrary point in aplane, M and N the feet of the perpendicularsdropped from 0 on the bisectors of theinterior and exterior angle A of a triangleABC; P and Qare defined in a similar mannerfor the angle B; Rand T for the angleC. Prove that the lines MN, PQ, and RT 77. Sec. 2. Selected Problems 77intersect at the same point or are parallel.48. Let 0 be the centre of the circle inscribedin a triangle ABC, A o, Bo, Cothe points of tangency of this circle with thesides BC, CA, AB, respectively. Taken onthe rays OAo,l OBo, OCo are points L, M,K, respectively, equidistant from the pointO. (a) Prove that the lines AL, BM, andCK meet in the same point. (b) Let AI'Bit C, be the projections of A, B, C, respectively,on an arbitrary line l passingthrough O. Prove that the lines AlL, BIM,and CIK are concurrent (that is, intersectat a common point).49. For the diagonals AD, BE, and CFof the hexagon ABCDEF inscribed in a circleto meet in the same point, it is necessaryandsufficient that the equality IAB IXI CD I I EF' = IBe I IDE I I FA Ibe fulfilled.50. Prove that: (a) the bisectors of the exteriorangles of a triangle intersect the extensionsof its opposite sides at three pointslying on the same straight line; (b) the tangentsdrawn from the vertices of the triangleto the circle circumscribed about it intersectits opposite sides at three collinearpoints.51. A circle intersects the side AB of atriangle ABC at points C1 and Cit the sideCA at points BI and BI , the side Be atpoints.d, andA.e Prove that if the li~esAAlt 78. 78 Problems in Plane GeometryBBt , and CCI meet in the same point, thenthe lines AA2 , BB2 , and CC2 also intersectat the same point.52. Taken on the sides AB, BC, and CAof a triangle ABC are points CI , AI' and81 Let C2 be the intersection point of thelines AB and AIBt , At the intersection pointof the lines BC and BtCt, B2 the intersectionpoint of the lines AC and AICI .Prove that if the lines AAt , BBl , andeCI meet in the same point, then thepoints AI' B2 , and C2 lie on a straightline.53. A straight line intersects the sides AB,BC, and the extension of the side AC ofa triangle ABC at points D, E, and F, re-_spectively. Prove that the midpoints of theline segments DC, AE, and BF lie on astraight line (Gaussian line).54. Given a triangle ABC. Let us definea point Al on the side BC in the followingway: Al is the midpoint of the side KLof a regular pentagon M KLNP whose verticesK and L lie on BC, and the vertices Mand N on AB and AC, respectively. Definedin a similar way on the sides AB andAC are points Ct and Bt Prove that thelines AA1 , BBl , and CCI intersect at thesame point. Gauss, Carl Friedrich (1777-1855). A Germanmatbematician. 79. Sec. 2.