KöMaL 2007 A434

Points $A$, $B$, $C$ lie inside the convex hexagon MNPQRS , such that the triangles ABC , NAM , PQB and CRS are similar. Let $X$, $Y$, $Z$ be the midpoints of the line segments $NP$, $QR$, $SM$, respectively, and let $G$, $K$, $I$ be the centroids of the triangles ABC , MPR and NQS , respectively. Prove that ($a$) If triangle ABC is equilateral then triangle GKI is equilateral; ($b$) Triangles ABC and XYZ are similar if and only if triangle ABC is equilateral.