Problem A. 796. (March 2021)

A. 796. Let \(\displaystyle ABCD\) be a cyclic quadrilateral. Let lines \(\displaystyle AB\) and \(\displaystyle CD\) intersect in \(\displaystyle P\), and lines \(\displaystyle BC\) and \(\displaystyle DA\) intersect in \(\displaystyle Q\). The feet of the perpendiculars from \(\displaystyle P\) to \(\displaystyle BC\) and \(\displaystyle DA\) are \(\displaystyle K\) and \(\displaystyle L\), and the feet of the perpendiculars from \(\displaystyle Q\) to \(\displaystyle AB\) and \(\displaystyle CD\) are \(\displaystyle M\) and \(\displaystyle N\). The midpoint of diagonal \(\displaystyle AC\) is \(\displaystyle F\).

Prove that the circumcircles of triangles \(\displaystyle FKN\) and \(\displaystyle FLM\), and the line \(\displaystyle PQ\) are concurrent.

Based on a problem by Ádám Péter Balogh, Szeged

(7 pont)

Deadline expired on April 12, 2021.

Statistics:

8 students sent a solution.
7 points:	Arató Zita, Balogh Ádám Péter, Bán-Szabó Áron, Diaconescu Tashi, Füredi Erik Benjámin, Török Ágoston.
6 points:	Sztranyák Gabriella.
2 points:	1 student.

Problems in Mathematics of KöMaL, March 2021