Pompeïu's Theorem states that for an equilateral $\Delta ABC$ and a point $P$ that is not on the circumcircle of the triangle, the lengths $PA, PB, PC$ form the sides of a triangle. Any such triangle is degenerate iff $P$ lies on the circumcircle.
The proof for this is simple enough, and can be obtained by a simple transformation. But, what about the converse? I was not able to find a proof for that anywhere.
EDIT: The converse states that if in a triangle $ABC$, for every point $P$ in its interior, $PB, PC, PA$ are the sides of a triangle, then $ABC$ is equilateral.