The triangle hiding inside every triangle

2026-05-13

Draw any triangle. Long and thin, lopsided, obtuse - it does not matter. Now, on each of its three sides, glue on an equilateral triangle pointing outward. Three identical tan kites sticking out, one per side.

Mark the centre of each kite. Connect those three centres with straight lines.

The blue triangle in the middle is equilateral. Its three sides are the same length, every time. Drag A, B, or C to any shape you like. Squash the outer triangle until it is barely a sliver. The blue triangle inside changes size, but its three sides stay locked at the same length as each other.

The outer triangle is allowed any shape. The inner one is not. The inner one is always perfect.

Why this is strange

It is one thing for a symmetric construction to spit out a symmetric result. If you start with an equilateral triangle, of course you get one back - the whole figure is symmetric from the first stroke.

But the outer triangle here has no symmetry at all. The three sides are different lengths. The three angles are different sizes. You have stuck on three equilaterals, yes, but they are three different sizes. There is no reason on the surface to expect their centres to land on a perfect equilateral.

And yet they do. Always.

Why it works

Each external equilateral has its centre two-thirds of the way along the line from a vertex to the apex - and that centre sits 30° off the side, at a fixed distance proportional to the side. So the three centres are produced from the three vertices by the same recipe: rotate by 30° outward, scale by 1/√3, average.

Three rotations by the same fixed angle, applied to three points, land on a triangle that is a rotated, scaled copy of the "average rotation" of the original. The fixed angle of 30° always combines with the law of cosines and the 1/√3 scale to give equal lengths. It is a one-line algebra problem in complex numbers - and a small miracle in pictures.

The Napoleon part

The theorem is named, perhaps unfairly, for Napoleon Bonaparte, who is supposed to have noticed it. He almost certainly did not prove it - it was floating around French mathematical circles in his time - but the attribution stuck, and the name is too good to drop.

There is a twin: if you build the equilaterals inward instead of outward, you get a second equilateral triangle. The two inner Napoleons differ in area by exactly the area of the original triangle.

A lopsided triangle, three equilateral kites stuck on its sides, three centres connected. The result has no business being regular. It is, every single time.