Question 52
Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $P$ be the intersection of $(BC)$ and the tangent to $\Gamma$ at $A$. Let $D$ and $E$ be the reflections of $B$ and $A$ respectively in $P$.
Let $\omega_{1}$ be the circumcircle of $DAC$ and $\omega_{2}$ that of $APB$. Let $F$ be the intersection of $\omega_{1}$ and $\omega_{2}$ other than $A$, and $G$ the intersection of $\omega_{1}$ with $(BF)$ other than $F$.
Prove that $(BC)$ and $(EG)$ are parallel.
Let $\omega_{1}$ be the circumcircle of $DAC$ and $\omega_{2}$ that of $APB$. Let $F$ be the intersection of $\omega_{1}$ and $\omega_{2}$ other than $A$, and $G$ the intersection of $\omega_{1}$ with $(BF)$ other than $F$.
Prove that $(BC)$ and $(EG)$ are parallel.