Assignment #4: Centers of a Triangle

By: Keith Schulte

Today's Topic: Centroid and
Medians

One center of a triangle is
the '**Centroid**', which is commonly
denoted by the letter '**G**', because
it represents the center of gravity of the triangle. It is created by the
intersection of the three **medians **of
a given triangle. By **definition:** a median of a triangle is a segment from a vertex to
the midpoint of the opposite side. In the figure below, we are given triangle
ABC. The points (A, B, and C) are
the vertices of the triangle. The midpoint of segment AB is point D. The
midpoint of segment BC is point E. The midpoint of segment AC is point F. The
line drawn from point C to point D is one of the three medians of the triangle.
The other two medians of the triangle are the segments AE and BF. The point
where the three medians intersect is the CENTROID, point G. If the triangle was
cut-out on a piece of cardboard, like the figure below, we should be able to
balance the triangle, by placing our finger underneath point G.

What other relationships can
we determine with regard to the Centroid and medians? If we create segments
connecting the midpoints of triangle ABC above, we would create a new triangle
inside of the original triangle with vertices at points D, E and F, see below

The new triangle, in red
above, is called the MEDIAL Triangle. We found above that point G is the center
of triangle ABC. By identifying the midpoints of each side of the triangle and
then creating a segment from a vertex to the midpoint on the opposite side of
the triangle. Is the relationship of G to the new triangle the same as it was
to the original triangle? That is, is G the center of gravity of the Medial
Triangle and the original triangle? Let's identify the midpoints of the Medial
Triangle and create segments between the vertices and the midpoints and see if
the intersection is the point G!

We see that point G is the
center of the medial triangle! If we made a medial triangle inside of this
medial triangle, would point G be its center also?

By creating the new triangle
with segments connecting the midpoints of the previous triangle, then finding the
midpoints of the new triangle and connecting the new midpoints to the vertices
of the new triangle, we find that point G is the center of the new triangle as
well. If we continued to make smaller and smaller medial triangles, we would
always find G to be the center of those triangles as well.