**Investigational Essay
Triangle Centers**

We will look at four different centers of triangles. These are the centroid, circumcenter, incenter and orthocenter. I have presented examples and definitions for each of the centers.

**Centroid**

Definition: The centroid of a triangle is the intersection of the three medians of the triangle. A median is the line that passes through the mid-point of one side of the triangle and the opposite vertex. Here is an example of a centroid.

I have used Geometer's sketch pad to look at the path of the centroid as our triangle moves from 0 degrees to 180 degrees. I have also constructed a right triangle that will rotate around the unit circle. I have traced the centroid of this triangle in hopes of seeing some characteristics of right triangle centroids.

**Circumcenter**

**Definition:** The circumcenter of a triangle is the intersection
of the three perpendicular bisectors of the triangle. A perpendicular bisectors
is the line that passes through the mid-point of one side of the triangle
and is perpendicular to that side. Here is an example of a circumcenter.

I have used Geometer's sketch pad to look at the path of the circumcenter as our triangle moves from 0 degrees to 180 degrees. I have also constructed a right triangle that will rotate around the unit circle. I have traced the circumcenter of this triangle in hopes of seeing some characteristics of right triangle

circumcenters.

**Incenter**

**Definition:** The Incenter of a triangle is the intersection of
the three angle bisectors of the triangle. A angle bisectors is the line
that passes through a vertex an divides that interior angle of the triangle
in half. Here is an example of an incenter.

I have used Geometer's sketch pad to look at the path of the incenter, as our triangle moves from 0 degrees to 180 degrees. I have also constructed a right triangle that will rotate around the unit circle. I have traced the circumcenter of this triangle in hopes of seeing some characteristics of right triangle incenters.

**Orthocenters**

**Definition:** The orthocenter of a triangle is the intersection
of the three altitudes of the triangle. An altitude is the line that passes
through a vertex and is perpendicular to the opposite side of the triangle.
Here is an example of an orthocenter.

I have used Geometer's sketch pad to look at the path of the orthocenter, as our triangle moves from 0 degrees to 180 degrees. I have also constructed a right triangle that will rotate around the unit circle. I have traced the orthocenter of this triangle in hopes of seeing some characteristics of right triangle orthocenters.