Question 23
Let $ABCD$ be a convex quadrilateral with $BC$ not parallel to $AD$. Suppose a point $E$ on side $BC$ makes both quadrilaterals $ABED$ and $AECD$ tangential (each has an incircle). Prove that there is a point $F$ on side $AD$ such that both $ABCF$ and $BCDF$ have incircles if and only if $AB\parallel CD$.