Let $ABC$ be a triangle with incenter $I$. Denote the midpoints of $AI, AC$ and $CI$ by $L, M$ and $N$ respectively. Point $D$ lies on segment $AM$ such that $BC= BD$. Let the incircle of triangle $ABD$ be tangent to $AD$ and $BD$ at $E$ and $F$ respectively.
Denote the circumcenter of triangle $AIC$ by $J$, and the circumcircle of triangle $JMD$ by $\omega$. Lines $MN$ and $JL$ meet $\omega$ again at $P$ and $Q$ respectively. Prove that $PQ, LN$ and $EF$ are concurrent.
See solution here.
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