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perpendicular wanted, AR=RQ, <CAP =<ABC, circle of diameter

Source: 2020 Croatia MO p7

February 16, 2021
geometryperpendicular

Problem Statement

A circle of diameter ABAB is given. There are points CC and D D on this circle, on different sides of the diameter such that holds AC<BCAC <BC or AC<ADAC<AD. The point PP lies on the segment BCBC and ∠CAP=∠ABC\angle CAP = \angle ABC. The perpendicular from the point CC to the line ABAB intersects the direction BDBD at the point QQ. Lines PQPQ and ADAD intersect at point RR, and the lines PQPQ and CDCD intersect at point TT. If AR=RQAR=RQ, prove that the lines ATAT and PQPQ are perpendicular.
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