In the figure below, equilateral
triangles ABC_{1}, BCA_{1}, and ACB_{1}
are constructed externally on the sides of a triangle ABC. Prove
that the centers A_{2}, B_{2}, and C_{2},
of those equilateral triangles form an equilateral triangle,
called the outer Napoleon triangle.
See
also:
Geometry Problem
246. Napoleon's Theorem I. External Equilateral triangles. Outer
triangle.

Apollonius for iPad: Napoleon's Theorem
I drew the geometric construction above using Apollonius for iPad.
Apollonius is an interactive geometry app (IGS, or Dynamic Geometry
Environment, DGE). It allows you to create geometric constructions, such as those that can be made using a ruler and a compass, and subsequently move parts of your constructions to explore the relationship between the geometric objects on the screen.
Apollonius is inspired by software such as GeoGebra, The Geometer's Sketchpad and Cabri,
but its interface is especially designed for touch screen.
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Problem
246 
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