Question 129
Let $I$ be the incenter of a triangle $ABC$. Write $L$, $M$, and $N$ for the midpoints of $AI$, $AC$, and $CI$, respectively. Assume that there is a point $D$ in the interior of the segment $AM$ such that $BD = BC$. The incircle of $\triangle ABD$ touches $AD$ and $BD$ at $E$ and $F$, respectively. Let $J$ be the circumcenter of $\triangle AIC$. Let $\omega$ denote the circumcircle of $\triangle DMJ$. The line $MN$ intersects $\omega$ at point $P(\neq M)$, and the line $JL$ intersects $\omega$ at point $Q(\neq J)$.
Prove that the three lines $EF$, $PQ$, and $LN$ pass through a common point.
Prove that the three lines $EF$, $PQ$, and $LN$ pass through a common point.