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.

CLICK HERE to see an example of the centroid of an obtuse triangle!!!
CLICK HERE to see an example of the centroid of an right triangle!!!
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.

CLICK HERE to see an example of the circumcenter of an acute triangle!!!
CLICK HERE to see an example of the circumcenter of an right triangle!!!

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.

CLICK HERE to see an example of the incenter of an acute triangle!!!
CLICK HERE to see an example of the incenter of an obtuse triangle!!!

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.

CLICK HERE to see an example of the orthocenter of an obtuse triangle!!!
CLICK HERE to see an example of the orthocenter of an right triangle!!!
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.