Question 71
Let $ABC$ be an acute triangle with $AB \lt AC$, $O$ be its circumcenter, and $M$ be the midpoint of arc $BC$ of triangle $ABC$'s circumcircle which does not include $A$. There is a point $D$ on the extension of side $AB$ beyond $B$ satisfying $BD = BM$, and there is a point $E$ on side $AC$ (except for end points) satisfying $CE = CM$. If the circumcircles of triangle $ABE$ and $ACD$ intersect at the point $X$ other than $A$, prove that the perpendicular bisector of segment $DE$ is tangent to the circumcircle of triangle $AOX$.