BALTIC-WAY 2014 Problem 12

Triangle $A B C$ is given. Let $M$ be the midpoint of the segment $A B$ and $T$ be the midpoint of the arc $B C$ not containing $A$ of the circumcircle of $A B C$. The point $K$ inside the triangle $A B C$ is such that $M A T K$ is an isosceles trapezoid with $A T \| M K$. Show that $A K=K C$.