Question 111
Let $I$ be the center of the inscribed circle of a tangential quadrilateral $ABCD$ (a quadrilateral circumscribed about a circle) with $\angle BAD \lt 90^\circ$. Points $E$ and $F$ lie on segments $BI$ and $DI$, respectively, such that $\angle EAF = \angle BAI$. Let $O$ be the center of the circumcircle of triangle $ABD$. $Q$ is the symmetric point of $O$ with respect to $BD$. If the points $B$, $D$, $Q$, and $C$ lie on the same circle, prove that $\angle ECF = \angle BAD$.
(Khulan Tumenbayar)
(Khulan Tumenbayar)