# Pompeiu's theorem 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. Proof of Pompeiu's theorem with Pompeiu triangle {displaystyle triangle PCP'} The proof is quick. Consider a rotation of 60° about the point B. Assume A maps to C, and P maps to P '. Then {displaystyle scriptstyle PB = P'B} , and {displaystyle scriptstyle angle PBP' = 60^{circ }} . Hence triangle PBP ' is equilateral and {displaystyle scriptstyle PP' = PB} . Then {displaystyle scriptstyle PA = P'C} . Thus, triangle PCP ' 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. 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. External links MathWorld's page on Pompeiu's Theorem Pompeiu's theorem at cut-the-knot.org Notes ^ Jump up to: a b c Jozsef Sandor: On the Geometry of Equilateral Triangles. Forum Geometricorum, Volume 5 (2005), pp. 107–117 ^ Jump up to: a b Titu Andreescu, Razvan Gelca: Mathematical Olympiad Challenges. Springer, 2008, ISBN 9780817646110, pp. 4-5 ^ 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) Categories: Elementary geometryTheorems about equilateral trianglesTheorems about triangles and circles

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