BALKAN-SL 2020 G4

Let $MAZN$ be an isosceles trapezium inscribed in a circle $(c)$ with centre $O$ . Assume that $MN$ is a diameter of $(c)$ and let $B$ be the midpoint of $AZ$ . Let $(\epsilon)$ be the perpendicular line on $AZ$ passing through $A$ . Let $C$ be a point on $(\epsilon)$ , let $E$ be the point of intersection of $CB$ with $(c)$ and assume that $AE$ is perpendicular to $CB$ . Let $D$ be the point of intersection of $CZ$ with $(c)$ and let $F$ be the antidiametric point of $D$ on $(c)$ . Let $P$ be the point of intersection of $FE$ and $CZ$ . Assume that the tangents of $(c)$ at the points $M$ and $Z$ meet the lines $AZ$ and $PA$ at the points $K$ and $T$ respectively. Prove that $OK$ is perpendicular to $TM$ .

Theoklitos Parayiou, Cyprus