Question 80
Let $\omega$ be the circumcircle of triangle $ABC$. Let $M,N$ be the midpoints of $AB,AC$, respectively, and $T$ the midpoint of the arc $BC$ not containing $A$. The circumcircles of $AMT,ANT$ meet the perpendicular bisectors of $AC,AB$ at $X,Y$, respectively; assume $X,Y$ lie inside $ABC$. Lines $MN,XY$ meet at $K$. Prove $KA=KT$.