Napoleon Variation Problem

Construct equilateral triangles BAD, BCE and CAF on the sides of a ΔABC, and reflect each of them respectively in sides BD, BE and CA. What do you notice about the triangle GHI formed by the centres of the reflected triangles A'BD, C'BE and F'CA?

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Napoleon Variation Problem

Challenge: Can you logically explain (prove) this result (which is a variation of Napoleon's Theorem)? If after spending some considerable time with it, and you're still stuck, go to Hint.

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Michael de Villiers, 25 June 2011.