a tour around the geometry of a cyclic quadrilateral
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7/28/2019 A tour around the geometry of a cyclic quadrilateral
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A Tour around the
Geometry of the
Cyclic Quadrilateral
School of Science, Mathematics and Technology Education
Faculty of Education
University of KwaZulu Natal
Durban
12 April 2013
Dr Chris Pritchardd
m2
m3m4
m1
ZY
X
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Alternate angles have equal sums
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Interior angle bisectors of any quadrilateral
meet at the vertices of a cyclic quadrilateral
360
360 180
180
EHG GFE a b c d
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Exterior angle
bisectors of an
arbitrary quadrilateralmeet at the vertices of
a cyclic quadrilateral
(and at the centres of
the escribed circles)
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Proof
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Perpendicular Bisectors
Perpendicular bisectorsare concurrent at the
circumcentre and divide
the cyclic quad into four
smaller cyclic quads
The circumcentres ofthe smaller cyclic
quads form the vertices
of another cyclic quad
This central cyclic quadis similar to the original.
In fact, STUVis an
enlargement ofABCD
(factor , centre O)
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Four Beer Mats Theorem
Four beer mats are
placed so that their
circumferences all
pass through aparticular point, P.
A dinner plate can
be positioned to fit
over the beer matsexactly.
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Four Beer Mats Theorem
Draw in four
equal diameters
to prove it!
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Five Beer Mats Theorem
Four beer mats are
placed so that their
circumferences all
pass through a
particular point, P.Common tangents
are drawn pairwise to
produce another
quadrilateral. An evenlarger dinner plate
can be positioned
over the quadrilateral
exactly.
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Revealing the Fifth Beer Mat
The fifth beer
mat has the
centres of the
first four on itscircumference.
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Miquels Six
Circle Theorem
ABCD is a cyclic quadrilateral.
Four circles, with centres outside
the quadrilateral, are drawn sothat the sides of the quadrilateral
are chords.
The second set of circle intersections,
E ,F, G, Hform the vertices of
another cyclic quadrilateral.
F
E
G
H
C
D
A
B
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Pairwise Incentres of a Cyclic Quadrilateral
ABCD is divided into four triangles, either side of a diagonal, twice.
Incircles are drawn.
The incentres form the vertices of a rectangle, JKLM.
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Pairwise Incentres of a Cyclic Kite
Now stir in some
bilateral symmetry
and the rectangle
becomes a square.
And a square has
equal diagonals.
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Pairs of Radii
Pairs of radii have
equal sum.
This is still true
if the symmetryis dropped, i.e.
for the general
cyclic quad.
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A Sangaku Cyclic Polygon Theorem (c. 1800)
F
F F
F
A
B
D
C
E
A
B
D
C
E
A
B
D
C
E
A
B
D
C
E
Divide a cyclic
polygon into
triangles arbitrarily.
The incircle radii
have the same total
length, regardless
of the configurationchosen.
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. . .AC BD AB CD BC DA If is any point on
the minor arc ,
then
B
ABC
BD AB CB
Ptolemys Theorem Van Schootens Theorem(special case of Ptolemys Theorem
whenACD is equilateral)
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Length of Diagonal of Regular
Pentagon of Unit Side
2 .1 1.1a a 2
1a a 2 1 0a a
1.618a
.
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Diagonal of Regular Heptagon of Unit Side
.
Applying Ptolemys Theorem toRVWXyields ab b a
.
2 1a b
3 2Eliminate to give 2 1 0.b a a a
Applying Ptolemys Theorem toRTWXyields
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3 2 2 1f a a a a
Need the root which is greater than 1.
Solution: 1.802, 2.247.a b
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Addition Formula for Sine:
Defining the Lengths of the Sides
Consider a circle with
unit diameter; then sin
cossin
cossin ( + )
+W
Z
OY
X
sin , sin ,
cos , cos ,
XY YZ
WX WZ
The full version of the Sine
Rule for triangle WXZis that:
sin=
sin=
sin= 2 = 1 if =
1
2
w= sinW
XZ = sin (+)
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Addition Formula for Sine:
Applying Ptolemys Theorem
By Ptolemys Theorem
. . .1.sin sin cos cos sin
sin sin cos cos sin
WY XZ XY WZ WX YZ
sin
cossin
cossin ( + )
+W
Z
OY
X
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cos
sin Z'
-
cossin
sin
sin ( - )
cos
WO
Y
X
90 -
sin (90 - )
sin (90 - ) sin
sin
sin (90 - ( - ))
90 -
90 - ( - )W
Z
OY
X
90 - ( + )90 - 90 -
Z'
sin (90 - )
sin (90 - )sin
sin (90 - ( - ))sin
WO
Y
X
Equivalent
Diagrams for other
Addition Laws
cos
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Double Angle Formulas
X'
sin
sin
cos
cos
sin 22
W
OY
X
1.sin2 sin cos cos sin
Let WY= 1
2 2
2 2
2 2
sin 90 1.sin 90 2 sin
cos cos 2 sin
cos 2 cos sin
90 -
X'
90 - 2sin
sin (90 - )
sin
W
O
Y
X
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Brahmagupta
(1) If the elements of two Pythagorean triples (k, l, m) and (K,L,M) arecombined to form products kM, mL, lMand mKrepresenting the
lengths of the sides of a quadrilateral, then the quadrilateral is cyclic.
A s a s b s c s d
(2) If the sides of a cyclic quadrilateral have lengths a, b, c, dand semi-
perimeters, then its area is given by:
(3) The lengths of the diagonals of a cyclic quadrilateral are given by:
ad bc ac bd xab cd
ab cd ac bd yad bc
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Brahmaguptas Trapezium
If the elements of two Pythagorean
triples (k, l, m) and (K,L,M) are
combined to form products kM, mL, lMand mKrepresenting the lengths of the
sides of a quadrilateral, then the
quadrilateral is cyclic.
(3, 4, 5) and (8, 15, 17) give a (40, 51, 68, 75) cyclic quadrilateral.
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Area and Diagonals of the
(40,51,69,75) Cyclic Quad
3234A s a s b s c s d
84
ad bc ac bd
x ab cd
77
ab cd ac bd y
ad bc
y
77
84x
421/2R
68
40
51
75
d
c
a
b
O
12
42
4
ab cd ac bd ad bcR
A
and by a formula of Paramesvara:
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Brahmaguptas Theorem and Corollary
For a cyclic quadrilateral withperpendicular diagonals meeting
at P(known as the anticentre), lines
drawn perpendicular to the four sides
through Pbisect the opposite sides.
P
T
M
B
DA
C
34
20251/2
371/2
68
40
51
75
O and the distance of thecircumcentre from each side is half
the length of the opposite side
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References 1 Alexander Bogomolnys Cut-the-knot website
Antonio Gutierrez website, Geometry Step by Step from the
Land of the Incas Eric Weissteins Wolfram MathWorld
H S M Coxeter & S L Greitzer, Geometry Revisited, MAA,1967.
Honsberger, R. More Mathematical Morsels. Washington, DC:
MAA (1991), 36-37; also Episodes in Nineteenth andTwentieth Century Euclidean Geometry. Washington, DC:MAA (1995), 35-40.
Hidetoshi Fukagawa & Daniel Pedoe, Japanese TempleGeometry Problems, Winnipeg: Charles Babbage ResearchFoundation, 1989.
Roger A Johnson, Modern Geometry: An Elementary Treatiseon the Geometry of the Triangle and the Circle. Boston, MA:Houghton Mifflin, 1929; also,Advanced Euclidean Geometry,New York: Dover, 1960.
Eli Maor, Trigonometric Delights, Princeton University Press,1998.
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References 2
Michael de VilliersSome Adventures in Euclidean
Geometry(2009)
Rethinking Proof with Geometers
Sketchpad, Key CurriculumPress, (2003)
Chris Pritchard (ed.)
The Changing Shape of Geometry,
Cambridge University Press / MAA (2003)
http://www.google.co.uk/url?sa=i&rct=j&q=michael+de+villiers+geometry&source=images&cd=&cad=rja&docid=N1EYEwVrrW6GRM&tbnid=ovTV-dpi6N9EsM:&ved=0CAUQjRw&url=http://www.amazon.com/Some-Adventures-In-Euclidean-Geometry/dp/images/0557102952&ei=AtJCUb8BwezSBdD8gJgI&psig=AFQjCNHXQD_bAhwkra659mEj6sbsFlyh6Q&ust=1363420032059339http://www.google.co.uk/url?sa=i&rct=j&q=chris+pritchard+geometry&source=images&cd=&cad=rja&docid=GEX30UBHsawffM&tbnid=RCpjeD_yeCxhjM:&ved=0CAUQjRw&url=http://www.mei.org.uk/?section=resources&page=books2&ei=cNFCUf6HLKbX0QXNsYDIBQ&bvm=bv.43828540,d.d2k&psig=AFQjCNFxxxQOf7SdJkzavwDtGc4ChIZxZA&ust=1363419815475540 -
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A Tour around the
Geometry of the
Cyclic Quadrilateral
School of Science, Mathematics and Technology Education
Faculty of Education
University of KwaZulu Natal
Durban
12 April 2013
Dr Chris Pritchard
chrispritchard2@aol.com
d
m2
m3m4
m1
ZY
X
W
P
mailto:chrispritchard2@aol.commailto:chrispritchard2@aol.commailto:chrispritchard2@aol.commailto:chrispritchard2@aol.com
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