MathNet 2007-part-03 — 185 problems (EN–VI) Question 36

Let $A$, $B$, $C$ be three distinct points on a unit circle. Let $G$ and $H$ be the centroid and the orthocenter of the triangle $ABC$, respectively. Let $F$ be the midpoint of the segment $GH$. Evaluate $|AF|^2 + |BF|^2 + |CF|^2$.