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PQ^3 /PN^2 = PS x RS /NS wanted, starting with 2 internally tangent circles

Source: Indonesia INAMO Shortlist 2008 G4

August 25, 2021
geometrytangent circles

Problem Statement

Given that two circles σ1\sigma_1 and σ2\sigma_2 internally tangent at NN so that σ2\sigma_2 is inside σ1\sigma_1. The points QQ and RR lies at σ1\sigma_1 and σ2\sigma_2, respectively, such that N,R,QN,R,Q are collinear. A line through QQ intersects σ2\sigma_2 at SS and intersects σ1\sigma_1 at OO. The line through NN and SS intersects σ1\sigma_1 at PP. Prove that PQ3PN2=PS⋅RSNS.\frac{PQ^3}{PN^2} = \frac{PS \cdot RS}{NS}.
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