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.

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 $$\scriptstyle\angle PBP'\ =\ 60^{\circ}$$. 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).

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{align}

a_{1,2}^2 &= \frac{1}{2}\left(PA^2 + PB^2 + PC^2 \pm 4\sqrt{3}\triangle_{(PA,PB,PC)}\right) \end{align}$$. 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 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.