How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle?

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How can construct three circles in a given triangle such that three internal tangent form an equilateral triangle?

**See also:**

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Draw three congruent equilateral triangles of an arbitrary size, with bases on the three sides centered at the incircle tangency points. Then the desired circles are inscribed in the kites delimited by the extended sides of these equilateral triangles as shown. Their common tangents form an equilateral triangle congruent to the other three.

This construction is most easily justified by working backwards. Find a solution among the one-parameter family with an equilateral triangle of the given size. Reflecting it over the common diameter of any two of the circles produces equilateral triangles with bases on the three sides. The quadrilaterals delimited by their extended sides are circumscribed about the circles, and so are kites by the equal opposite angles. That means any two of them are equidistant from the vertex between them, which means their bases are centered at the incircle tangency points.

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