Question 59
Let $ABC$ be a triangle with circumcircle $\omega$. Let $\ell_{B}$ and $\ell_{C}$ be parallel lines through $B$ and $C$ respectively. Let $D$ be the intersection of $\omega$ and $\ell_{B}$ other than $B$, and $E$ the intersection of $\omega$ and $\ell_{C}$ other than $C$.
Suppose $\ell_{C}$ and $(AD)$ meet at $F$, while $\ell_{B}$ and $(AE)$ meet at $G$. Let $O, O_{1}$ and $O_{2}$ be the circumcentres of $ABC$, $ADG$ and $AEF$ respectively. Let $P$ be the circumcentre of $OO_{1}O_{2}$.
Prove that $(OP)$ is parallel to $\ell_{B}$ and $\ell_{C}$.
Suppose $\ell_{C}$ and $(AD)$ meet at $F$, while $\ell_{B}$ and $(AE)$ meet at $G$. Let $O, O_{1}$ and $O_{2}$ be the circumcentres of $ABC$, $ADG$ and $AEF$ respectively. Let $P$ be the circumcentre of $OO_{1}O_{2}$.
Prove that $(OP)$ is parallel to $\ell_{B}$ and $\ell_{C}$.