Question 163
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 $G$, $E$ ($E \neq A$), respectively; the line $d_{2}$ cuts $O_{1}$ and $O_{2}$ at $F$, $H$ ($F \neq A$), 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 $K$, $L$ ($K, L \neq B$), respectively. Let $N$ be the intersection of $O_{1}K$ and $O_{2}L$. Prove that the circle $(NLK)$ is tangent to $AB$.
Let $J$ be the intersection of $EH$ and $FG$. The line $BJ$ cuts $O_{1}$, $O_{2}$ at $K$, $L$ ($K, L \neq B$), respectively. Let $N$ be the intersection of $O_{1}K$ and $O_{2}L$. Prove that the circle $(NLK)$ is tangent to $AB$.