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2022 TMO Geometry

Source: 2022 Taiwan Mathematics Olympiad

February 19, 2022
geometry

Problem Statement

Let JJ be the AA-excenter of an acute triangle ABCABC. Let XX, YY be two points on the circumcircle of the triangle ACJACJ such that JX‾=JY‾<JC‾\overline{JX} = \overline{JY} < \overline{JC}. Let PP be a point lies on XYXY such that PBPB is tangent to the circumcircle of the triangle ABCABC. Let QQ be a point lies on the circumcircle of the triangle BXYBXY such that BQBQ is parallel to ACAC. Prove that ∠BAP=∠QAC\angle BAP = \angle QAC.
Proposed by Li4.
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