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 60ð about the point B. Assume A maps to C, and P maps to Pâ². Then , and . Hence triangle PBPâ² is equilateral and . Then . Thus, triangle PCPâ² has sides equal to PA, PB, and PC and the proof by construction is complete (see drawing).
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 â PA, PB, and PC two equilateral triangles (the larger and the smaller) with sides a<sub>1</sub> and a<sub>2</sub> are defined:
The symbol â³ 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 MöbiusâÂÂPompeiu theorem.