Question 95
Let $ABC$ be a triangle inscribed in the circle $C$ with center $O$ and radius $1$. For any point $M \in C \setminus \{A, B, C\}$, we denote $s(M) = OH_1^2 + OH_2^2 + OH_3^2$, where $H_1$, $H_2$, and $H_3$ are the orthocenters of triangles $MAB$, $MBC$, and $MCA$, respectively.
a) Prove that if triangle $ABC$ is equilateral, then $s(M) = 6$, for any $M \in C \setminus \{A, B, C\}$.
b) Prove that if there exist three distinct points $M_1, M_2, M_3 \in C \setminus \{A, B, C\}$ such that $s(M_1) = s(M_2) = s(M_3)$, then triangle $ABC$ is equilateral.
a) Prove that if triangle $ABC$ is equilateral, then $s(M) = 6$, for any $M \in C \setminus \{A, B, C\}$.
b) Prove that if there exist three distinct points $M_1, M_2, M_3 \in C \setminus \{A, B, C\}$ such that $s(M_1) = s(M_2) = s(M_3)$, then triangle $ABC$ is equilateral.