Vijetnam 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 $A(x_0,y_0)$ ($y_0\ne 0$), then the coordinates of the orthocenter $H$ satisfy the following equations $$\{\begin{array}{l}{x}_H=x_0\\ (x_H+a)(a-x_0)-y_0{y}_H=0\end{array}$$ 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 $$3a^2-3x_0^2+y_0^2=0⟺\frac{x_0^2}{a^2}-\frac{y_0^2}{3a^2}=1(y_0\ne 0).$$ 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 $C$.