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2017 preRMO p27, circle lies on interior of another circle, radius wanted

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August 20, 2019
geometrycircleradiuscircles

Problem Statement

Let Ω1\Omega_1 be a circle with centre OO and let ABAB be diameter of Ω1\Omega_1. Let PP be a point on the segment OBOB different from OO. Suppose another circle Ω2\Omega_2 with centre PP lies in the interior of Ω1\Omega_1. Tangents are drawn from AA and BB to the circle Ω2\Omega_2 intersecting Ω1\Omega_1 again at A1A_1 and B1 respectively such that A1A_1 and B1B_1 are on the opposite sides of ABAB. Given that A1B=5,AB1=15A_1 B = 5, AB_1 = 15 and OP=10OP = 10, find the radius of Ω1\Omega_1.
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