Question 165
Let $ABC$ be a triangle. Let $D$, $E$, $F$ be points on the sides $BC$, $AC$, $AB$ respectively, such that
$$
DE \parallel AB,\ DF \parallel AC \text{ and } \frac{BD}{DC} = \frac{AB^2}{AC^2}.
$$
The circumcircle of the triangle $AEF$ meets $AD$ at $R$ again and meets the line passing through $A$ and tangent to the circumcircle of the triangle $ABC$ at $S$ again. Let the line $EF$ intersect $BC$ at $L$ and $SR$ at $T$. Prove that $SR$ bisects $AB$ if and only if $BS$ bisects $TL$.
$$
DE \parallel AB,\ DF \parallel AC \text{ and } \frac{BD}{DC} = \frac{AB^2}{AC^2}.
$$
The circumcircle of the triangle $AEF$ meets $AD$ at $R$ again and meets the line passing through $A$ and tangent to the circumcircle of the triangle $ABC$ at $S$ again. Let the line $EF$ intersect $BC$ at $L$ and $SR$ at $T$. Prove that $SR$ bisects $AB$ if and only if $BS$ bisects $TL$.