MathDB
TB = TQ wanted, 5 circumcircles related

Source: 2019 Balkan MO Shortlist G5 - BMO

November 19, 2020
geometrycircumcircleequal segments

Problem Statement

Let ABCABC (BC>ACBC > AC) be an acute triangle with circumcircle kk centered at OO. The tangent to kk at CC intersects the line ABAB at the point DD. The circumcircles of triangles BCD,OCDBCD, OCD and AOBAOB intersect the ray CACA (beyond AA) at the points Q,PQ, P and KK, respectively, such that P∈(AK)P \in (AK) and K∈(PQ)K \in (PQ). The line PDPD intersects the circumcircle of triangle BKQBKQ at the point TT, so that PP and TT are in different halfplanes with respect to BQBQ. Prove that TB=TQTB = TQ.
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