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PR bisects <MPN, internally tangent circles

Source: Netherlands - Dutch MO 1984 p1

December 25, 2022
geometryangle bisectortangent circles

Problem Statement

The circles C1C_1 and C2C_2 with radii r1r_1 and r2r_2 touch the line pp at the point PP. C1C_1 lies entirely within C2C_2. Line q⊥pq \perp p intersects pp at SS and touches C1C_1 at RR. qq intersects C2C_2 at MM and NN, where NN is between RR and SS. a) Prove that line PRPR bisects angle ∠MPN\angle MPN. b) Calculate the ratio r1:r2r_1 : r_2 if line PNPN bisects angle ∠RPS\angle RPS.
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