Problem

Source: Turkey 2021 IMO TST Problem 5

Tags: Turkey, geometry, perpendicular bisector, menelaus theorem



In a non isoceles triangle $ABC$, let the perpendicular bisector of $[BC]$ intersect $(ABC)$ at $M$ and $N$ respectively. Let the midpoints of $[AM]$ and $[AN]$ be $K$ and $L$ respectively. Let $(ABK)$ and $(ABL)$ intersect $AC$ again at $D$ and $E$ respectively, let $(ACK)$ and $(ACL)$ intersect $AB$ again at $F$ and $G$ respectively. Prove that the lines $DF$, $EG$ and $MN$ are concurrent.