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Equilateral Action (1)!

The applet below demonstrates a few properties of an equilateral triangle. Feel free to move the white points (and the brown point that will soon appear) anywhere you'd like! The first property (involving the equilateral triangle's angles) should be more than obvious. Yet how would you put the phenomena you dynamically observe (in the last half of the slider action) into words? And how could you possibly prove this property that holds true for all equilateral triangles?