Nemotron-IMO-Bench 2026 G16

On the sides $AB, BC,CD, DE, EF, FA$ of a convex hexagon $ABCDEF$ we built equilateral triangles, either all outside the hexagon or all towards its interior. Let $M,N,P,Q,R,S$ be the centers of these triangles and let $X,Y,Z$ be the midpoints of $MQ, NR, PS$. Prove that $XYZ$ is an equilateral triangle.