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Equilateral Action (3)!

Creation of this applet was inspired by this tweet from Alexander Bogomolny (@CutTheKnotMath) You will eventually see an equilateral triangle appear with circles that are tangent to this triangle and to each other. The segments (soon to be constructed) will also be tangent to the circles they touch. Be sure to move the BIG INNER WHITE POINT on the equilateral triangle's incircle anywhere you'd like (at any time!) Can you formally prove what this applet informally illustrates?