2010 JBMO Shortlist G1

Consider a triangle ${ABC}$ with${\angle ACB=90^{\circ}.}$ Let ${F}$ be the foot of the altitude from ${C}$. Circle ${\omega}$ touches the line segment ${FB}$at point ${P,}$  the altitude ${CF}$at point ${Q}$ and the circumcircle of ${ABC}$at point ${R.}$ Prove that points ${A,Q,R}$ are collinear and ${AP=AC}$.

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