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.
![](image1.gif)
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.
CLICK HERE if you would like
access to the GSP file I used to create the picture below!!!
![](image23.gif)
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.
![](image4.gif)
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.
CLICK HERE if you would
like access to the GSP file I used to create the picture below!!!
![](image22.gif)
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.
![](image5.gif)
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.
CLICK HERE if you would like
access to the GSP file I used to create the picture below!!!
![](image25.gif)
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.
![](image7.gif)
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.
CLICK HERE if you would
like access to the GSP file I used to create the picture below!!!
![](image26.gif)