Question 54
Let $ABCD$ be a cyclic quadrilateral whose sides $BC$ and $AD$ are not parallel. Let $E$ be a point inside the circumcircle of $ABCD$ which is on the opposite side of the line $AB$ with respect to the point $C$. The lines $DE$ and $AB$ meet at $F$. Let $G$ be a point inside $ABCD$ and also on the line which is tangent to the circumcircle of triangle $AEF$ at $E$. If
$$
\angle GAD = \angle BAE \text{ and } \angle GCB + \angle GAB = \angle EAD + \angle AGD + \angle ABE
$$
then show that the lines $BC$, $AD$ and $EG$ are concurrent.
$$
\angle GAD = \angle BAE \text{ and } \angle GCB + \angle GAB = \angle EAD + \angle AGD + \angle ABE
$$
then show that the lines $BC$, $AD$ and $EG$ are concurrent.