 Mathematical and Physical Journal
for High Schools
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# 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