Question 67
Let $ABC$ be a triangle and $\Gamma$ a circle through $A$. Suppose $\Gamma$ meets segments $[AB]$ and $[AC]$ again at $D$ and $E$ respectively, and meets $[BC]$ at $F$ and $G$ with $F$ between $B$ and $G$. Let $T$ be the intersection of the tangent at $F$ to the circumcircle of $BDF$ and the tangent at $G$ to the circumcircle of $CEG$.
Prove that if $A$ and $T$ are distinct, then $(AT)$ and $(BC)$ are parallel.
Prove that if $A$ and $T$ are distinct, then $(AT)$ and $(BC)$ are parallel.