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ST // BC , circumcircle, point on angle bisector

Source: 2015 VMEO IV Juniors 12.2 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

September 15, 2021
geometryparallelangle bisector

Problem Statement

Given a triangle ABCABC inscribed in circle (O)(O) and let PP be a point on the interior angle bisector of BACBAC. PBPB, PCPC cut CACA, ABAB at E,FE,F respectively. Let EFEF meet (O)(O) at M,NM,N. The line that is perpendicular to PMPM, PNPN at M,NM,N respectively intersect (O)(O) at S,TS, T different from M,NM,N. Prove that ST∥BCST \parallel BC.
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