Problem

Source: Iran RMM TST 2019,day2 p6

Tags: geometry



Let $ABCD $ be cyclic quadrilateral with circumcircle $\omega $ and $M $ be any point on $\omega $. Let $E $ and $F $ be the intersection of $AB,CD $ and $AD,BC $ respectively. $ME $ intersects lines $AD,BC $ at $P,Q $ and similarly $MF$ intersects lines $AB,CD $ at $R,S $. Let the lines $PS $ and $RQ $ meet at $X $. Prove that as $M $ varies over $\omega $ $MX $ passes through fixed point. Proposed by Mehdi Etesami Fard