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Fermat point, orthocenter

Source: 2022 Viet Nam math olympiad for high school students D1 P7

March 20, 2023
geometry

Problem Statement

Let ABCABC be a triangle with ∠A,∠B,∠C<120∘\angle A,\angle B,\angle C <120^{\circ}, TT is its Fermat-Torricelli point. Let Ha,Hb,HcH_a, H_b, H_c be the orthocenter of triangles TBC,TCA,TABTBC, TCA, TAB, respectively. a) Prove that: TT is the centroid of the △HaHbHc\triangle H_aH_bH_c. b) Denote D,E,FD, E, F respectively by the intersections of HcHbH_cH_b and the segment BCBC, HcHaH_cH_a and the segment CACA, HaHbH_aH_b and the segment ABAB. Prove that: the triangle DEFDEF is equilateral. c) Prove that: the lines passing through D,E,FD, E, F and are respectively perpendicular to BC,CA,ABBC, CA, AB are concurrent at a point. Let that point be SS. d) Prove that: TSTS is parallel to the Euler line of the triangle ABCABC.
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