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Another easy geometry from Taiwan TST

Source: 2021 Taiwan TST Round 2 Independent Study 2-G

April 8, 2021
geometrycircumcirclegeometric transformationreflection

Problem Statement

Let ABCABC be a triangle with circumcircle Γ\Gamma, and points EE and FF are chosen from sides CACA, ABAB, respectively. Let the circumcircle of triangle AEFAEF and Γ\Gamma intersect again at point XX. Let the circumcircles of triangle ABEABE and ACFACF intersect again at point KK. Line AKAK intersect with Γ\Gamma again at point MM other than AA, and NN be the reflection point of MM with respect to line BCBC. Let XNXN intersect with Γ\Gamma again at point SS other that XX.
Prove that SMSM is parallel to BCBC.
Proposed by Ming Hsiao
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