Problem

Source: Balkan MO 2016, Problem 2

Tags: geometry, cyclic quadrilateral, Nine Point Circle, Balkan Mathematics Olympiad, Balkan, bir tiyinga qimmat masala



Let $ABCD$ be a cyclic quadrilateral with $AB<CD$. The diagonals intersect at the point $F$ and lines $AD$ and $BC$ intersect at the point $E$. Let $K$ and $L$ be the orthogonal projections of $F$ onto lines $AD$ and $BC$ respectively, and let $M$, $S$ and $T$ be the midpoints of $EF$, $CF$ and $DF$ respectively. Prove that the second intersection point of the circumcircles of triangles $MKT$ and $MLS$ lies on the segment $CD$. (Greece - Silouanos Brazitikos)