Pompeiu's theorem
Pompeiu's theorem is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is simple, but not classical. It states the following:
The proof is quick. Consider a rotation of 60° about the point B. Assume A maps to C, and P maps to P
Further investigations reveal that if P is not in the interior of the triangle, but rather on the circumcircle, then PA, PB, PC form a degenerate triangle, with the largest being equal to the sum of the others; this observation is also known as Van Schooten's theorem.
Generally, by the point P and the lengths to the vertices of the equilateral triangle - PA, PB, and PC two equilateral triangles with sides and are defined:
The symbol △ denotes the area of the triangle whose sides have lengths PA, PB, PC.
Pompeiu published the theorem in 1936; however August Ferdinand Möbius had already published a more general theorem about four points in the Euclidean plane in 1852. In this paper Möbius also derived the statement of Pompeiu's theorem explicitly as a special case of his more general theorem. For this reason, the theorem is also known as the Möbius-Pompeiu theorem.