RO-TST 2013 Problem 6

The vertices of two acute-angled triangles lie on the same circle. The Euler circle (nine-point circle) of one of the triangles passes through the midpoints of two sides of the other triangle. Prove that the triangles have the same Euler circle.

EDIT by pohoatza (in concordance with Luis' PS): Alternate/initial version Let $ABC$ be a triangle with circumcenter $\Gamma$ and nine-point center $\gamma$ . Let $X$ be a point on $\Gamma$ and let $Y$ , $Z$ be on $\Gamma$ so that the midpoints of segments $XY$ and $XZ$ are on $\gamma$ . Prove that the midpoint of $YZ$ is on $\gamma$ .