ISL 2003 G5

Let $ABC$ be an isosceles triangle with $AC=BC$ , whose incentre is $I$ . Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$ . The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$ , respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$ , respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$ .

Proposed by Hojoo Lee