**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 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 *C*. Assume *A* maps to *B*, and *P* maps to *P* '. Then we have
P
C
=
P
′
C
, and
∠
P
C
P
′
=
60
∘
. Hence triangle *PCP* ' is equilateral and
P
P
′
=
P
C
. It is obvious that
P
A
=
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.