Proof that the medians of any triangle meet at one point. Proof #2 Using algebra and analytic geometry.

We have already demonstrated, using classical geometry, that the centroid lies at a point which divides each median in a ratio of 2:1. We will use the information about where the medians intersect to help us prove the theorem using analytic geometry. 2 1

First, we calculate the coordinates of the midpoints of each side of the triangle. Here are the coordinates of the vertices of our triangle. ? ? ?

Next, we need to find the coordinates of the points that split each median into a ratio of 2:1, then verify that these points are identical. We consult a book on analytic geometry, and find that the formula for the point that divides a line into a ratio of 2:1 is Let’s put that formula aside for now.

Next, we need to find the coordinates of the points that split each median into a ratio of 2:1, then verify that these points are identical. First, we insert the coordinates of the endpoints of our first line into our formula: =

Next, we need to find the coordinates of the points that split each median into a ratio of 2:1, then verify that these points are identical. Next, we insert the coordinates of the endpoints of our second line. =

Next, we need to find the coordinates of the points that split each median into a ratio of 2:1, then verify that these points are identical. Finally, we insert the coordinates of the endpoints of our third line. =

Hence, the point at a ratio of 2:1 on each median is the same, and is the centroid of the triangle.