SHARYGIN 2017 Problem 6

Let $ABCD$ be a convex quadrilateral with $AC = BD = AD$; $E$ and $F$ the midpoints of $AB$ and $CD$ respectively; $O$ the common point of the diagonals. Prove that $EF$ passes through the touching points of the incircle of triangle $AOD$ with $AO$ and $OD$.