Mathematics
Education
EMAT 6680, Professor Wilson
Exploration 4, The Centroid of a Triangle by Ursula Kirk
The Centroid of a
triangle is the point of intersection of the three medians. A median of a
triangle is the segment from the vertex to the midpoint of the other side.
In this investigation I
will show using vectors that the centroid is located at
.
The Medians and the
Centroid
Assume we have any
triangle with vertices A, B and C. We find the midpoints of each of the sides
of the triangle which are m1, m2 and m3. Then, we construct a line segment from
each of the vertices A, B and C to each of our midpoints m1, m2 and m3. The
point where these three lines intersect is the centroid. The centroid of a
triangle is of special significance because it is the center of mass of a
system obtained by placing equal masses at the given points.
Finding the centroid
with Vectors
In order to find the
midpoint in a triangle formed by vectors we have to use vector addition.
As we can see in the
picture, the midpoint of the sum of the two vectors u and w can
be easily found by constructing the vector sum u + v and then finding
the midpoint of the resultant. So this midpoint is 
as shown in the picture.
We can use the same
technique to allocate the midpoints for the three sides of our triangle as we
can see below.
So we have our
midpoints are ,
and 
.
Using these midpoints, we construct the medians by joining the points with the
vertices of the triangle that are vectors
,
,
.
The
point of intersection of these three medians is the centroid.
Next, I want to prove
that the three medians do indeed intersect at point which is the centroid of our triangle.
Looking at our picture,
it looks like the medians meet at the point 2/3 of the way from 
to 
, 2/3 of the way from to and 2/3 of the way from to .
Therefore the point 2/3
of the way from to is at
=
Similarly, we can
calculate the other two medians and the results will also be .
Therefore the centroid
is at
