Assignment 4

Investigation #5

Heather Robinson

EMAT 6680, Fall 2003

The triangle is an interesting geometric figure and exploring the many “centers” of a triangle is an extensive investigation. In this investigation, we have explored the centroid, circumcenter, orthocenter, and incenter of a triangle. Definitions for these centers are as follows…

The centroid is defined as
the point of concurrency for the *medians* in a triangle. The *medians* are the segments in a
triangle created by connecting the midpoint of a side to the vertex of the
angle opposite of the side. The medians
are always on the triangle’s interior; therefore, the centroid
will always be on the triangle’s interior.

The circumcenter is the
concurrency of the *perpendicular bisectors*. *Perpendicular bisectors* cut a side into two equal
halves.

The orthocenter can be
described as the concurrency of the *altitudes*. An *altitude* can be drawn from a vertex so that it is
perpendicular to the opposite side.
Altitudes may be inside or outside of the triangle; therefore, the orthocenter may be inside or outside of the
triangle.

The incenter is the
concurrency of the *angle bisectors*.
*Angle bisectors* cut an angle into two equal halves.

The following facts hold true for the centroid, circumcenter, orthocenter, and incenter…

1. In an equilateral triangle, these four points are the same.

2. The circumcenter, centroid, and orthocenter are always collinear.

EXAMPLE 1

EXAMPLE 2