NL-NWO 1997 Problem 5

Given is a triangle $ABC$ and a point $K$ within the triangle. The point $K$ is mirrored in the sides of the triangle: $P , Q$ and $R$ are the mirrorings of $K$ in $AB , BC$ and $CA$ , respectively . $M$ is the center of the circle passing through the vertices of triangle $PQR$ . $M$ is mirrored again in the sides of triangle $ABC$ : $P', Q'$ and $R'$ are the mirror of $M$ in $AB$ respectively, $BC$ and $CA$ . a. Prove that $K$ is the center of the circle passing through the vertices of triangle $P'Q'R'$ . b. Where should you choose $K$ within triangle $ABC$ so that $M$ and $K$ coincide? Prove your answer.