AoPS - Problem

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Problem

Source: 2021 Saudi Arabia JBMO TST 1.2

Tags: geometry, hexagon, Cyclic

parmenides51

04.09.2021 18:44

In a circle $O$ , there are six points, $A$ , $B$ , $C$ , $D$ , $E$ , $F$ in a counterclockwise order such that $BD \perp CF$ , and $CF$ , $BE$ , $AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$ , and the perpendicular from $D$ to $CE$ be $N$ . Prove that $AE \parallel MN$ .

Jalil_Huseynov

04.09.2021 20:32

Let $K=BD\cap CF$ and $AD,BE,CF$ are concurrent at $G$ . We have $AD$ is antiparallel to $BC$ and since $BMKC$ is cyclic, $MK$ is antiparallel to $BC$ $\implies$ $AD||MK$ ,similarly $BE||NK$ . So that is enough to prove $\triangle AGE\sim \triangle MKN$ . $\frac{AG}{MK}=\frac{GC}{KC}=\frac{GE}{NK}$ and we have $\angle AGE=\angle MKN$ from parallel lines. So we are done.