Centroid of triangle proof?

Question

Prove that the centroid of a plate triangle of uniform thickness, and the center of mass of 3 identical bodies, one at each vertice of the triangle, are the same point? Any short cuts to this besides looking up the formulas? Why are both the intersection of the medians of the triangle?

Answer 1

let A’C’ be parallel to AC with distance dx << 1 between AC and A’C’, and A’ belonging to AB, and C’ belonging to CB; the mass center D of trapezium AA’C’C being inside is close to the midpoints of AC and A’C’; and BD is a median; thence mass center is the intersection point of medians;
The same with equal masses; the mass center of masses A and C is in the midpoint D; the mass center of mass B and effective mass D is point E, being ED = 1/3 of BD; according to property of medians E is their cross point.
A smarter proof lies in definition that mass center of ABC is positioned on axis of minimal moment of inertia.