We consider the triangle below.
If we draw a segment from each vertex to the opposite
side's midpoint, we find that in the above
case, a common point of intersection occurs in the
center of the triangle. This is defined as the
Centroid, G. You'll notice that all three segments
are concurrent to the Centroid and G appears to be
the center of the triangle....
Things to try with the a triangle and a given
Question 1 Is it always the case that these lines from
each vertex to the midpoint to the opposite
sides are concurrent to G ?
Question 2 Would a triangle's moment or center of mass
(if uniformly massive & dense) be the
location of the Centroid?
Question 3 If we can draw a circle inside a triangle
that is tangent to all three sides, what is the relaionship
between the radius of the triangle and the centroid?
(hint: construct the circle then explore the triangles
that can exist with it have each side tangent to the circle)
To try the above problems out and any others using
Geometry Sketchpad 4.0 with the tool below
Click here to explore
the triangle with a GSP 4.1 centroid tool
Solutions to above questions
3. Answers vary
Proofs and other implications are left to the reader and are
classic explorations for high school students.