Selection tests for the Balkan Mathematical Olympiad 2013

To prove that $C$, $F$, $I$, and $J$ are concyclic, it is equivalent to prove that $\angle CFI=\angle CJI$ or $\angle CFI+\angle CJI=180^{\circ }$, depending on the configuration. We will present here the proof for one configuration. The proof for the other configuration is similar.



Because $AFCH$ is cyclic, we have $\angle CFI=\angle CAH$.

Because $\angle FEA=\angle FGA=90^{\circ }$, the quadrilateral $AFEG$ is cyclic, and therefore $\angle CAH=\angle EFG$.

It remains to prove that $\angle EFG=\angle DJG$, which is equivalent to proving that quadrilateral $DGFJ$ is cyclic.

But $\angle EGF=\angle EAF$ since $AFEG$ is cyclic. On the other hand, because $AFCD$ is cyclic, we deduce that $\angle EAF=\angle CDF$. Therefore, $\angle EGF=\angle CDF$, which proves that $DGFJ$ is cyclic.