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Orthogonal Circles

Source: 2019 Korea Winter Program Practice Test 1 Problem 2

January 12, 2019
geometrycircumcircle

Problem Statement

ω1,ω2\omega_1,\omega_2 are orthogonal circles, and their intersections are P,P′P,P'. Another circle ω3\omega_3 meets ω1\omega_1 at Q,Q′Q,Q', and ω2\omega_2 at R,R′R,R'. (The points Q,R,Q′,R′Q,R,Q',R' are in clockwise order.) Suppose P′RP'R and PR′PR' meet at SS, and let TT be the circumcenter of △PQR\triangle PQR. Prove that T,Q,ST,Q,S are collinear if and only if O1,S,O3O_1,S,O_3 are collinear. (OiO_i is the center of ωi\omega_i for i=1,2,3i=1,2,3.)
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