Question 167
Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles intersecting at $M$ and $N$. Let $A$ and $B$ be the points of contact of these two circles with their common external tangent closest to $M$. Let $C$ and $D$ be the reflections of $A$ and $B$ respectively through $M$, and let $E$ and $F$ be the intersection points (other than $M$) of the circumcircle of $MCD$ with $\Gamma_{1}$ and $\Gamma_{2}$ respectively. Show that the circumcircles of $MEF$ and $NEF$ have the same radius.