, 2007

Choose orthogonal Cartesian coordinate system $Oxy$ where $O$ is the midpoint of the segment $BC$ and $Oy$ is the line $BC$. Denote by $2a > 0$ the length of the segment $BC$. The coordinates of the vertices $B$ and $C$ are $B(-a, 0)$ and $C(a, 0)$. Suppose that $A$ has the coordinates $A(x_0, y_0)$ ($y_0 \neq 0$), then the coordinates of the orthocenter $H$ satisfy the following equations
$$\begin{cases} x_H = x_0 \\ (x_H + a)(a - x_0) - y_0 y_H = 0 \end{cases}$$
hence $H\left(x_0, \frac{a^2 - x_0^2}{y_0}\right)$. The centroid $G$ has the coordinates $\left(\frac{x_0}{3}, \frac{y_0}{3}\right)$. The midpoint $K$ of $HG$ has the coordinates $\left(\frac{2x_0}{3}, \frac{3a^2 - 3x_0^2 + y_0^2}{6y_0}\right)$. The point $K$ belongs to the line $BC$ if and only if
$$3a^2 - 3x_0^2 + y_0^2 = 0 \iff \frac{x_0^2}{a^2} - \frac{y_0^2}{3a^2} = 1 \quad (y_0 \neq 0).$$
Thus, the locus of $A$ is the hyperbola $\frac{x_0^2}{a^2} - \frac{y_0^2}{3a^2} = 1$ except the two points $B$ and $C$.