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Geometry with two circles

Source: Serbia TST 2024, P5

May 18, 2024
geometry

Problem Statement

The circles k1,k2k_1, k_2, centered at O1,O2O_1, O_2, meet at two points, one of which is AA. Let P,QP, Q lie on AO1,AO2AO_1, AO_2, respectively, so that PQ∥O1O2PQ \parallel O_1O_2. The tangents from PP to k2k_2 touch it at X,YX, Y and the tangents from QQ to k1k_1 touch it at Z,TZ, T. Show that X,Y,Z,TX, Y, Z, T are collinear or concyclic.
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