Pompeiu's theorem
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Pompeiu's theorem is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is quite simple, but not classical. It states the following:
- Given an equilateral triangle ABC in the plane, and a point P in the triangle ABC, the lengths PA, PB, and PC form the sides of a nondegenerate triangle.
The proof is quick. Consider a rotation of 
about the point C. Assume A maps to B, and B maps to B'. Then we have PC = P'C, and 
. Hence triangle PCP' is equilateral and PP' = PC. It is obvious that PA = P'B. Thus, triangle PBP' has sides equal to PA, PB, and PC and the proof by construction is complete.
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.
