Given a triangle $ABC$; in its plane, a line $e$ and a point $P$ on it.

$1^{\circ}$. Find a point $X$ on the line $e$ such that

$$
\overline{X A}^{2}+\overline{X B}^{2}+\overline{X C}^{2} \text { is minimized }
$$

$2^{\circ}$. Find the geometric locus of this $X$ point when the line $e$ rotates around the fixed point $P$.
I. Solution $1^{\circ}$. Let us choose the line $e$ as the abscissa axis; the ordinate axis perpendicular to it can be arbitrary. The coordinates of the vertices of the given triangle are respectively: $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)$. The coordinates of the sought point $X$ are $(x, 0)$. The given

$$
\begin{gathered}
\overline{X A}^{2}+\overline{X B}^{2}+\overline{X C}^{2} \text { can be written as: } \\
f(x)=\left(x-x_{1}\right)^{2}+y_{1}^{2}+\left(x-x_{2}\right)^{2}+y_{2}^{2}+\left(x-x_{3}\right)^{2}+y_{3}^{2} \\
=3 x^{2}-2\left(x_{1}+x_{2}+x_{3}\right) x+C
\end{gathered}
$$

where $C$ is a constant composed of known values. The quadratic function $f(x)$ takes its minimum value when

$$
x=\frac{x_{1}+x_{2}+x_{3}}{3}
$$

this is the abscissa of the sought point; since the abscissa of the centroid $(S)$ of the triangle is the same, the sought point $X$ is the foot of the perpendicular dropped from the centroid to the line $e$.

$2^{\circ}$. The angle $S X P$ is a right angle for any position of the line $e$; since the distance $S P$ is constant, the geometric place of the point $X$ is the circle drawn over $S P$ as a diameter.

(Friedmann Bernát.)

II. Solution Let $X^{\prime}$ be an arbitrary point on the line $e$; then:

$$
{\overline{A X^{\prime}}}^{2}+{\overline{B X^{\prime}}}^{2}+{\overline{C X^{\prime}}}^{2}=\overline{S A}^{2}+\overline{S B}^{2}+\overline{S C}^{2}+3{\overline{S X^{\prime}}}^{2}
$$

The expression in question is minimized when $S X^{\prime}$ is minimized.

Thus, the sought point $X$ is obtained by dropping a perpendicular from $S$ to $e$.

The problem was also solved by: Devecis M., Kertész L., Kornis Ö., Roth M., Sasvári G., Spitzer Ö., Szabó I., Szabó K.