Napoleon's theorem

Dependencies: Rotation properties (rotations preserve distance and angle), Central ∠, arc, chord are pairwise equivalent & Central angle is twice the inscribed angle on the same arc (central angle of an equilateral triangle's circumscribed circle = $120^\circ$), Three equal angles ⇒ equilateral (three $60^\circ$ angles $\Rightarrow$ equilateral), Composition = isometry (the composition of two rotations is again a rotation).

Statement

Let $\triangle ABC$ be an arbitrary non-degenerate triangle. On its three sides, construct outward equilateral triangles

where $A'$ and $A$ are on opposite sides of $BC$, $B'$ and $B$ are on opposite sides of $CA$, and $C'$ and $C$ are on opposite sides of $AB$. Let $O_A$, $O_B$, $O_C$ be the centers of these three equilateral triangles (i.e. centroid / circumcenter). Then the three centers form an equilateral triangle:

This is Napoleon's theorem: on the three sides build equilaterals outward, and their centers form an equilateral. Replacing "outward" by "inward" yields the inner Napoleon triangle — likewise equilateral, with a parallel conclusion.