The line segments joining the vertices of triangle ABC to the midpoints of the opposite edges are called medians and are concurrent at P.
Prove that P splits each median in the ratio 2:1.
Consider the following diagram.
As C" is the midpoint of AB, AC"=C"B, so the area of triangle APC" = area of triangle C"PB = A1. Similarly area BPA" = area A"PC = A2 and area CPB" = area B"PA = A3.
As BB" bisects the area of the triangle, A3+2A1 = A3+2A2; hence A1=A2.
Let us consider the way that PB splits triangle ABA": triangles ABP and PBA" have the same altitude, and as area ABP = 2 area PBA", it follows that AP = 2(PA"), proving that P splits the median AA" in the ratio 2:1.
Prove that AA", BB", and CC" are concurrent.