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A familiar segment sum condition in a cyclic quadrilateral

Source: 4th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane"- Junior D2 P5/ Senior D2 P4

December 21, 2022
geometrycyclic quadrilateralsegment sumangle bisectorCircumcentercircumcircle

Problem Statement

Let ABCDABCD be a cyclic quadrilateral such that AB=AD+BCAB = AD + BC and CD<ABCD < AB. The diagonals ACAC and BDBD intersect at PP, while the lines ADAD and BCBC intersect at QQ. The angle bisector of ∠APB\angle APB meets ABAB at TT. Show that the circumcenter of the triangle CTDCTD lies on the circumcircle of the triangle CQDCQD.
Proposed by Nikola Velov
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