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.''Titu Andreescu, Razvan Gelca: ''Mathematical Olympiad Challenges''. Springer, 2008, , pp
4-5
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The proof is quick. Consider a rotation of 60° about the point ''B''. Assume ''A'' maps to ''C'', and ''P'' maps to ''P'' '. Then $\scriptstyle PB\ =\ P'B$, and $\scriptstyle\angle PBP'\ =\ 60^$. Hence triangle ''PBP'' ' is equilateral and $\scriptstyle PP'\ =\ PB$. Then $\scriptstyle PA\ =\ P'C$. Thus, triangle ''PCP'' ' has sides equal to ''PA'', ''PB'', and ''PC'' and the proof by construction is complete (see drawing).Jozsef Sandor
''On the Geometry of Equilateral Triangles''
Forum Geometricorum, Volume 5 (2005), pp. 107–117

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 ( the larger and the smaller) with sides $a_1$ and $a_2$ are defined:
:$\begin a_^2 &= \frac\left(PA^2 + PB^2 + PC^2 \pm 4\sqrt\triangle_\right) \end{align}$.
The symbol △ denotes the area of the triangle whose sides have lengths ''PA'', ''PB'', ''PC''.Mamuka Meskhishvili
'' Two Non-Congruent Regular Polygons Having Vertices at the Same Distances from the Point''
International Journal of Geometry, Volume 12 (2023), pp. 35–45

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''.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

External links

Pompeiu's theorem
at cut-the-knot.org

Notes

Elementary geometry
Theorems about equilateral triangles
Theorems about triangles and circles
Articles containing proofs