Problem

Source: 2015 Singapore Seniors Round 2 p5

Tags: geometry, circles, collinear, Tangents



Let $A$ be a point on the circle $\omega$ centred at $B$ and $\Gamma$ a circle centred at $A$. For $i=1,2,3$, a chord $P_iQ_i$ of $\omega$ is tangent to $\Gamma$ at $S_i$ and another chord $P_iR_i$ of $\omega$ is perpendicular to $AB$ at $M_i$. Let $Q_iT_i$ be the other tangent from $Q_i$ to $\Gamma$ at $T_i$ and $N_i$ be the intersection of $AQ_i$ with $M_iT_i$. Prove that $N_1,N_2,N_3$ are collinear.