Triangle ABC has equilateral triagles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. Experimentation with the interactive diagram leads to the conjecture that PQR is an equilateral triangle. This can be proved using vectors or complex numbers.

In the following w = e ^{p i/3} so that 1 +w+ w ² = 0. Also multiplying a complex number by w rotates it by 60 degrees. Referring to the given diagram let A, B, ¼ be represented by the complex numbers a, b,¼. The third vertex of the triangle with base AB is represented by the complex number b+w( a- b). Therefore P is represented by p where

Similarly

To show that PQR is equilateral it is sufficient to show that w( q- p) = r- p and this follows using simple algebra and 1 +w+ w ² = 0.