GeoProof — Napoleon's Theorem explorer

The three centers form an equilateral triangle

Napoleon's Theorem

Take any triangle at all. On each of its three sides, build an equilateral triangle pointing outward. Mark the center of each one. Then:

the three centers form an equilateral triangle

This outer Napoleon triangle is perfectly equilateral no matter how lopsided the original triangle is — drag the vertices and watch its three sides stay equal.

Inner version too

If you instead build the equilateral triangles pointing inward, their centers also form an equilateral triangle — the inner Napoleon triangle. Switch the "Build inward" toggle to see it.

A bonus relationship

Both Napoleon triangles share the same center as the original triangle's centroid. And the difference between the outer and inner Napoleon triangles' areas equals the area of the original triangle.

Why "center"?

For an equilateral triangle the centroid, circumcenter, and incenter all coincide, so "the center" is unambiguous — it sits one-third of the way up from each side, at distance side ÷ (2√3) from the midpoint.

A bit of history

The result is traditionally attributed to Napoleon Bonaparte, who was a keen amateur mathematician, though historians doubt he actually discovered it. It first appears in print in the early 19th century.