So, a friend of mine came up with this unorthodox proof of the centroid's existence so I figured I could share it here so that someone can confirm that it's a fine one. I think it is correct, but I haven't seen anything like it. I will write it unrigorously because I'm a high school student and I'm not that good at writing formal proofs.
Let K, L, M be the midpoints of the sides of triangle ABC. We assume that the medians of ABC don't intersect in a point, so their intersections form a triangle of finite area T. Then the medians of ABC are also the medians of KLM (I know this needs to be proved. However, I also know how to prove it correctly, so let us assume this is true). KLM has 1/4 of the area of ABC. We can repeat this procedure over and over again so that we get a triangle smaller than T. Now, T must be inside the triangle because it contains the intersections of the medians, which are inside the triangle, so that T must be smaller than the other triangle. This is a contradiction, so the three medians intersect at one point.
Thanks for the help!
