$E$ and $F$ are points respectively lying on $AB$ and $CA$ such that $B, C, E, F$ are concyclic. $BF \cap CE = K$. $KM \perp CA$ and $KN \perp AB$ $(M \in CA$ and $N \in AB)$ and $L$ is the midpoint of $AK$. $I$ and $J$ are points lying respectively on $EF$ and $BC$ such that $\angle AKI = \angle KJI$. $(JML)$ and $(JNL)$ respectively cuts $CK$ and $BK$ at $P$ and $Q$. Prove that $\overline{J, N, P}$ and $\overline{J, M, Q}$. Note that $I$ is the midpoint of $EF$.
It can be seen, but not proven, that $J$ is the midpoint of $BC$.
We might be able to establish that $MNPQ$ is a parallelogram, perhaps even an isosceles parallelogram. But I am one of "us".