Medians & Equal Areas!

Recall if 2 triangles have equal bases and equal heights, they have equal areas. Since this is so, we can informally conclude that the 3 medians of any triangle split this triangle up into 6 smaller triangles of equal area. Feel free to move the triangle's white vertices anywhere you'd like at any time. Again, this applet displays an informal illustration of this fact using transformational geometry techniques. Yet how could you formally prove what this applet informally illustrates?

Quick Demo (BGM: Andy Hunter)