CN-TST 2020 Problem 2

Given an isosceles triangle $\triangle ABC$ , $AB=AC$ . A line passes through $M$ , the midpoint of $BC$ , and intersects segment $AB$ and ray $CA$ at $D$ and $E$ , respectively. Let $F$ be a point of $ME$ such that $EF=DM$ , and $K$ be a point on $MD$ . Let $\Gamma_1$ be the circle passes through $B,D,K$ and $\Gamma_2$ be the circle passes through $C,E,K$ . $\Gamma_1$ and $\Gamma_2$ intersect again at $L \neq K$ . Let $\omega_1$ and $\omega_2$ be the circumcircle of $\triangle LDE$ and $\triangle LKM$ . Prove that, if $\omega_1$ and $\omega_2$ are symmetric wrt $L$ , then $BF$ is perpendicular to $BC$ .