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Problem 1461

AIME late

Geometry Difficulty 5.9 Prove it Team Selection Tests for JBMO · Saudi Arabia · 2018

Let $ABC$ be a triangle inscribed in circle $(O)$ with incenter $I$ . The lines $IB$ and $IC$ intersect $(O)$ again at $J$ and $L$ . Circumcircle $(\omega)$ of triangle $IBC$ meets $CA$ , $AB$ again at $E$ , $F$ . Prove that $EL$ and $FJ$ intersect on $(\omega)$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Denote $X$ as the other intersection of $(IJL)$ and $(\omega)$ . We shall prove that $X$ is the intersection of $EL$ and $FJ$ .

By the cyclic quadrilateral, we have
$\angle LXI = \angle LJI = \angle LJB = \angle LCB = \angle LCA = 180^\circ - \angle EXI.$
Hence $L$ , $X$ , $E$ are collinear, which means $X$ belongs to $EL$ .

Similarly, $JF$ passes through $X$ . These finish our proof.

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