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Internal common tangents and angle condition

Source: China TST 3 2018 Day 1 Q1

March 27, 2018
geometry

Problem Statement

Let ω1,ω2\omega_1,\omega_2 be two non-intersecting circles, with circumcenters O1,O2O_1,O_2 respectively, and radii r1,r2r_1,r_2 respectively where r1<r2r_1 < r_2. Let AB,XYAB,XY be the two internal common tangents of ω1,ω2\omega_1,\omega_2, where A,XA,X lie on ω1\omega_1, B,YB,Y lie on ω2\omega_2. The circle with diameter ABAB meets ω1,ω2\omega_1,\omega_2 at PP and QQ respectively. If ∠AO1P+∠BO2Q=180∘,\angle AO_1P+\angle BO_2Q=180^{\circ}, find the value of PXQY\frac{PX}{QY} (in terms of r1,r2r_1,r_2).
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