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:

Given an equilateral triangle ABC in the plane, and a point P in the plane of the triangle ABC, the lengths PA, PB, and PC form the sides of a (maybe, degenerate) triangle.[1][2]
Proof of Pompeiu's theorem with Pompeiu triangle

The proof is quick. Consider a rotation of 60° about the point B. Assume A maps to C, and P maps to P '. Then , and . Hence triangle PBP ' is equilateral and . Then . Thus, triangle PCP ' has sides equal to PA, PB, and PC and the proof by construction is complete (see drawing).[1][2]

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.[1]

Pompeiu published the theorem in 1936, however August Ferdinand Möbius had published a more general theorem about four points in the Euclidean plane already 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.[3]

Notes
  1. Jozsef Sandor: On the Geometry of Equilateral Triangles. Forum Geometricorum, Volume 5 (2005), pp. 107–117
  2. Titu Andreescu, Razvan Gelca: Mathematical Olympiad Challenges. Springer, 2008, ISBN 9780817646110, pp. 4-5
  3. D. MITRINOVIĆ, J. PEČARIĆ, J., V. VOLENEC: History, Variations and Generalizations of the Möbius-Neuberg theorem and the Möbius-Ponpeiu. Bulletin Mathématique De La Société Des Sciences Mathématiques De La République Socialiste De Roumanie, 31 (79), no. 1, 1987, pp. 25–38 (JSTOR)
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