This Demonstration gives a proof of Pompeïu's theorem: If is a point in the plane of the equilateral triangle , then there exists a triangle with side lengths , , and unless lies on the circumcircle of the triangle , when the triangle is degenerate.
We rotate the equilateral triangle around the point by an angle of . This gives the triangle . As and , is equilateral, so . Also, and , so we have constructed whose sides have the same lengths as , , and .