Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.
Problem
Source: India Postal Coaching 2014 Set 2 Problem 2 & Sharygin 2014
Tags: geometry, circumcircle, symmetry, incenter, power of a point, radical axis, geometry unsolved