Triangles AA*C, A*BC are equal and triangle A'CB' is similar to the former two by a similarity with fixed ratio. This leads to a trivial proof. Notice that ABC and A'B'C' are perspective, the intersection point of the lines joining opposite vertices of them lying on the Kiepert hyperbola of ABC.

Notice that AA*, BB*, CC* are the

Notice also that the Napoleon triangle persists even when the triangle is degenerate (its vertices are collinear).

The three images above show the triangles formed by the centers of equilaterals erected on the sides of a basic triangle (ABC) but with orientations different from the previous. The first has two outside and one inside. The second has two inside and one outside and the third has all three inside (

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