Question 88
Let $ABC$ be an acute-angled triangle inscribed in a circle with center $O$ and radius $R$, and let $H$ be the orthocenter of triangle $ABC$. Let $A_1$ be a point on the side $BC$ such that $HA_1 + A_1O = R$. Similarly, define points $B_1$ on $CA$ and $C_1$ on $AB$. If $\overrightarrow{AA_1} + \overrightarrow{BB_1} + \overrightarrow{CC_1} = \overrightarrow{0}$, prove that triangle $ABC$ is equilateral.
Alexandru Cărnaru
Alexandru Cărnaru