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Two circles

Source: Mexican Math Olympiad 2012 - problem 1

December 1, 2013
trigonometrygeometry unsolvedgeometry

Problem Statement

Let C1\mathcal{C}_1 be a circumference with center OO, PP a point on it and ℓ\ell the line tangent to C1\mathcal{C}_1 at PP. Consider a point QQ on ℓ\ell different from PP, and let C2\mathcal{C}_2 be the circumference passing through OO, PP and QQ. Segment OQOQ cuts C1\mathcal{C}_1 at SS and line PSPS cuts C2\mathcal{C}_2 at a point RR diffferent from PP. If r1r_1 and r2r_2 are the radii of C1\mathcal{C}_1 and C2\mathcal{C}_2 respectively, Prove PSSR=r1r2.\frac{PS}{SR} = \frac{r_1}{r_2}.
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