Školjka

%V0 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 $ABC$. comment (According to my team leader, last year some of the countries wanted a geometry question that was even easier than this...that explains IMO 2003/4...) [Note by Darij: This was also Problem 6 of the German pre-TST 2004, written in December 03.] Edited by Orl.