SAUDI ARABIAN MATHEMATICAL COMPETITIONS

Given two circles $O_1$ and $O_2$ intersect at $A$ and $B$. Let $d_1$ and $d_2$ be two lines through $A$ and be symmetric with respect to $AB$. The line $d_1$ cuts $O_1$ and $O_2$ at $G$, $E$ ($E\ne A$), respectively; the line $d_2$ cuts $O_1$ and $O_2$ at $F$, $H$ ($F\ne A$), respectively; such that $E$ is between $A$, $G$ and $F$ is between $A$, $H$. Let $J$ be the intersection of $EH$ and $FG$. The line $BJ$ cuts $O_1$, $O_2$ at $K$, $L$ ($K,L\ne B$), respectively. Let $N$ be the intersection of $O_{1}K$ and $O_{2}L$. Prove that the circle $(NLK)$ is tangent to $AB$.