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2 circumcircles concurrent with angle bisector

Source: 2014 Oral Moscow Geometry Olympiad grades 8-9 p4

August 9, 2019
geometryperpendicular bisectorcircumcircleconcurrencyconcurrentangle bisector

Problem Statement

In triangle ABCABC, the perpendicular bisectors of sides ABAB and BCBC intersect side ACAC at points PP and QQ, respectively, with point PP lying on the segment AQAQ. Prove that the circumscribed circles of the triangles PBCPBC and QBAQBA intersect on the bisector of the angle PBQPBQ.