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collinear wanted, 2 circles and exsimilicenter center

Source: 2016 Thailand October Camp 1.5

October 15, 2020
geometrycollinearcircles

Problem Statement

Let ω1,ω2\omega_1, \omega_2 be two circles with different radii, and let HH be the exsimilicenter of the two circles. A point X outside both circles is given. The tangents from XX to ω1\omega_1 touch ω1\omega_1 at P,QP, Q, and the tangents from XX to ω2\omega_2 touch ω2\omega_2 at R,SR, S. If PRPR passes through HH and is not a common tangent line of ω1,ω2\omega_1, \omega_2, prove that QSQS also passes through HH.
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