Question 151
Let $ABC$ be an isosceles triangle with $|AC| = |BC|$ and incircle center $I$. Let $P$ be a point on the circumcircle of triangle $AIB$, which lies in triangle $ABC$. The straight lines through $P$, parallel to $CA$ and $CB$, intersect $AB$ into $D$ and $E$. The line parallel to $AB$ through $P$ intersects $CA$ and $CB$ into $F$ and $G$. Show that the two lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$.