Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC = 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} = 3$. Find the locus of the point $ A$ so that $ F\in BC$.
Problem
Source: Moldova National MO 2008, 12 Grade, Problem 7 (Day 2)
Tags: geometry, conics, hyperbola, geometry proposed