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Concentric circles in parallelogram

Source: 2024 Israel TST Test 6 P3

March 20, 2024
geometryparallelogramconcentric circlesTSTperpendicular bisectorcircumcircle

Problem Statement

Let ABCDABCD be a parallelogram. Let ω1\omega_1 be the circle passing through DD tangent to ABAB at AA. Let ω2\omega_2 be the circle passing through AA tangent to CDCD at DD. The tangents from BB to ω1\omega_1 touch it at AA and PP. The tangents from CC to ω2\omega_2 touch it at DD and QQ. Lines APAP and DQDQ intersect at XX. The perpendicular bisector of BCBC intersects ADAD at RR.
Show that the circumcircles of triangles △PQX\triangle PQX, △BCR\triangle BCR are concentric.
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