OME 1995 Problem 6

Let $C$ be a variable interior point of a fixed segment $AB$ . Equilateral triangles $ACB'$ and $CBA'$ are constructed on the same side and $ABC'$ on the other side of the line $AB$ . (a) Prove that the lines $AA' ,BB'$ , and $CC'$ meet at some point $P$ . (b) Find the locus of $P$ as $C$ varies. (c) Prove that the centers $A'' ,B'' ,C''$ of the three triangles form an equilateral triangle. (d) Prove that $A'' ,B'',C''$ , and $P$ lie on a circle.