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Symmedian involving geometry

Source: 2023 Turkey NMO 2nd Round P6

December 21, 2023
geometrycircumcirclesymmedian

Problem Statement

On a triangle ABCABC, points DD, EE, FF are given on the segments BCBC, ACAC, ABAB respectively such that DE∥ABDE \parallel AB, DF∥ACDF \parallel AC and BDDC=AB2AC2\frac{BD}{DC}=\frac{AB^2}{AC^2} holds. Let the circumcircle of AEFAEF meet ADAD at RR and the line that is tangent to the circumcircle of ABCABC at AA at SS again. Let the line EFEF intersect BCBC at LL and SRSR at TT. Prove that SRSR bisects ABAB if and only if BSBS bisects TLTL.
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