Joined Common Chords of Napoleon's Circumcircles

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29 July 2013, Created with GeoGebra

An Equilateral Triangle in Napoleon's Circumcircles

Let $ABC',$ $BCA',$ and $CAB'$ be Napoleon's triangles constructed on the sides of $\Delta ABC.$ Choose point $D$ on the circumcircle $ABC',$ pass it through $A$ to the intersection $F$ with $C(CAB')$ and through $E$ to the intersection $E$ with $C(BCA').$

An equilateral triangle in Napoleon's circumcircles

Triangles $DEF$ is equilateral.

Hint

The problem is very simple; it submits to chasing inscribed angles.

Solution

The solution is outlined in an old variant of this problem.

Acknowledgment

The problem described on this page stems from an observation of Hirotaka Ebisui.

Napoleon's Theorem

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