SHARYGIN 2023 Problem 20

Let a point $D$ lie on the median $AM$ of a triangle $ABC$ . The tangents to the circumcircle of triangle $BDC$ at points $B$ and $C$ meet at point $K$ . Prove that $DD'$ is parallel to $AK$ , where $D'$ is isogonally conjugated to $D$ with respect to $ABC$ .