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weird sum for the orthocenters

Source: Romanian District Olympiad, grade 10, problem 2

March 10, 2024
geometrycomplex number geometry

Problem Statement

Let ABCABC be a triangle inscribed in the circle C(O,1)\mathcal{C}(O,1). Denote by s(M)=OH12+OH22+OH32,s(M)=OH_1^2+OH_2^2+OH_3^2, (∀)M∈C∖{A,B,C},(\forall) M \in\mathcal{C}\setminus \left\{A,B,C\right\}, where H1,H2,H3,H_1,H_2,H_3, are the orthocenters of the triangles MAB, MBCMAB,~MBC and MCA.MCA. a)a) Prove that if ABCABC is equilateral,, then s(M)=6,(∀)M∈C∖{A,B,C},s(M)=6,(\forall) M \in\mathcal{C}\setminus \left\{A,B,C\right\}, b)b) Prove that if there exist three distinct points M1,M2,M3∈C∖{A,B,C}M_1,M_2,M_3\in\mathcal{C}\setminus \left\{A,B,C\right\} such that s(M1)=s(M_1)=s(M2)s(M_2)=s(M3),=s(M_3), then ABCABC is equilateral..
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