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