Question 97
Given a triangle $ABC$, a circle centered at some point $O$ meets the segments $BC, CA, AB$ in the pairs of points $X$ and $X'$, $Y$ and $Y'$, $Z$ and $Z'$, respectively, labeled in circular order: $X, X', Y, Y', Z, Z'$. Let $M$ be the Miquel point of the triangle $XYZ$ (i.e., the point of concurrence of the circles $AYZ, BZX, CXY$), and let $M'$ be that of the triangle $X'Y'Z'$. Prove that the segments $OM$ and $OM'$ have equal lengths.
