Tool Box Concurrent, Median, ReflectionCentroid, Circumcenter, Altitude

Triangles – Points of Concurrency

IncenterIncenter

Located at intersection of the angle bisectors.

Recall - An angle bisector cuts an angle in half.There are three angles in a triangle; therefore, there are three angle bisectors in a

triangle

Their point of intersection is called the incenter. FYI – FACT!A triangle's incircle - the largest circle that will fit inside the triangle with the

incenter as the circle’s center!

Mrs. Kelley – SHOW WEBSITE!!!

Angle Bisector

FYI – FACT!A triangle's incircle - the largest circle that will fit inside the triangle with the incenter as the circle’s

center!

More angle bisector facts!• Any point ON the angle bisector is equidistant

from the sides of the angle itself! • Remember! Distance is the PERPENDICULAR

line from the point to the side of the angle (ray) see below

Distance – ANY point ON the bisector is equal distan

Centroid

CentroidLocated at intersection of the medians

It is the point of balance for the triangle; it is the center of gravity!

The median is a line segment that connects the vertex to the middle of a side opposite the vertex.

“cuts” the side in half – NOT the vertex (except in special cases like equilateral triangle and vertex angle of an isosceles triangle)

There are three sides and three vertexes in a triangle, therefore, there are three medians.

Picture of Triangle Medians meeting the Centroid!

Centroid Theorem “2/3 Thm”

• The centroid is LOCATED 2/3 of the way from the VERTEX to the opposite side.

• AO = 2/3 AE• BO = 2/3 BF• CO = 2/3 CD

• REMEMBER TO START AT VERTEX!!!

Orthocenter

Orthocenter (prefix ‘ortho’ means ‘right’)Located at intersection of the altitudes or HEIGHT OF THE TRIANGLE!

An altitude of a triangle is it’s height. A Triangle has three heights!The altitude is a line segment that connects a vertex and is

perpendicular to the opposite side (DOES NOT HAVE TO BE IN THE MIDDLE of either side or angle but is now PERPENDICULAR to the side!!)

There are three vertexes and sides in a triangle, therefore, there are three altitudes.

Facts about Perpendicular!

• Perpendicular• Box• 90 degrees

Circumcenter

CircumcenterLocated at intersection of the perpendicular bisectors of the sides

A perpendicular bisector is a line that cuts a side in half and is also perpendicular to the side.

There are three sides in a triangle therefore, there are three perpendicular bisectors

The point of intersection is called the circumcenter.

Circumcenter Picture with details

EH AC AE = ECFG CD CF = FDJB AD AB = BD G

H

J

Circumcenter FACTS!• The circumcenter is equidistant from the vertices of the

triangle!

• Also – centerof circle OUTSIDE the triangle that touches all vertices (as opposed to incenter)

Euler Line – Named after famous 18th century Swiss mathematician – Leonard Euler(1707-1783)

In any triangle, the centroid, circumcenter and orthocenter always lie on a straight line, called the Euler line.

Useful Websites – click on the link to be directed to website (accessed on October 10, 2012)

mathopenref.com http://www.mathopenref.com/trianglecenters.html

Concurrence summary http://www.regentsprep.org/Regents/math/geometry/GC3/LConcurrence.htm