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AP is perpendicular to QR iff AB=AC or 2BC^2=AB^2+AC^2

Source: 2022 Viet Nam math olympiad for high school students D1 P8

March 20, 2023

Problem Statement

Given the triangle ABCABC with TT is its Fermat–Torricelli point. Let (Na)(N_a) be the circumcircle of △TBC\triangle TBC. Choose a point XX on (Na)(N_a) such that TXTX is perpendicular to BCBC. The segment BCBC intersects (TNaX)(TN_aX) at DD. Similar definition of points Y,Z,E,FY, Z, E, F. The reflection lines of the Euler line of △ABC\triangle ABC wrt BC,CA,ABBC, CA, AB intersect XD,YE,ZFXD, YE, ZF at P,Q,RP, Q, R, respectively.
Prove that: APAP is perpendicular to QRQR if and only if AB=ACAB = AC or 2BC2=AB2+AC22BC^2 = AB^2 + AC^2.
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