The coordinates of the vertices of a triangle are $A(5 ; 0), B(0 ; 5)$, and $C(-4 ;-3)$. Determine the point $P$ on the circumcircle of the triangle for which the expression $P A^{2}+P B^{2}+P C^{2}$ is minimized.
It is easy to see from the coordinates that $C$ is at the same distance from the origin $O$ as $A$ and $B$, so the equation of the circumcircle $k$ is

$$
x^{2}+y^{2}=25
$$

Let the coordinates of $P$ be $(u, v)$ and those of $A$ be $\left(x_{A}, y_{A}\right)$. Then, based on (1),

$$
\begin{gathered}
P A^{2}=\left(u-x_{A}\right)^{2}+\left(v-y_{A}\right)^{2}=\left(u^{2}+v^{2}\right)-2 x_{A} u-2 y_{A} v+\left(x_{A}^{2}+y_{A}^{2}\right)= \\
=2 \cdot 25-2 x_{A} u-2 y_{A} v
\end{gathered}
$$

since the coordinates of $P$ and $A$ both satisfy (1).

![](https://cdn.mathpix.com/cropped/2024_05_02_edb563de8f3b5fa21fc2g-1.jpg?height=602&width=672&top_left_y=618&top_left_x=716)

Similarly, $P B^{2}$ and $P C^{2}$ can be expressed using the coordinates of $B\left(x_{B}, y_{B}\right)$ and $C\left(x_{C}, y_{C}\right)$, and thus the sum of squares to be examined is:

$$
f(P)=150-2\left(x_{A}+x_{B}+x_{C}\right) u-2\left(y_{A}+y_{B}+y_{C}\right) v
$$

is minimal if and only if the expression

$$
\left(x_{A}+x_{B}+x_{C}\right) u+\left(y_{A}+y_{B}+y_{C}\right) v
$$

which, in our case, is $u+2 v$, is maximal, subject to the condition that $(u, v)$ lies on $k$, i.e., $u^{2}+v^{2}=25$.

Now, the value of the expression $u+2 v$ for a given $c$ is constant for the points on the line

$$
\begin{aligned}
x+2 y & =c, \quad \text { or equivalently } \\
y & =-\frac{1}{2} x+\frac{c}{2}
\end{aligned}
$$

The line $e$ passes through the point $\left(0 ; \frac{c}{2}\right)$ on the $y$-axis, and increasing $c$ unambiguously means that this point moves upward along the axis. However, we can only increase $c$ until the line (3) has a common point with the circle (1), and the largest possible value of $c$ is obtained when the line (3) is tangent to $k$. The point of tangency $T$ is the point where the perpendicular from the center of the circle to $e$ intersects $k$. The equation of the perpendicular is $y=2 x$, and its intersection points with $k$ are:

$$
T_{1}(\sqrt{5}, 2 \sqrt{5}), \quad T_{2}(-\sqrt{5},-2 \sqrt{5})
$$

and $u+2 v$ clearly takes its maximum value at $T_{1}$, which is $(c=5 \sqrt{5})$, so (2) is minimal here, and this is the desired point $P$. (In $T_{2}$, however, $f(P)=f\left(T_{2}\right)$ takes its maximum value.)

Remarks. 1. The expression (2) can be written as $3 x_{S} u+3 y_{S} v$, where $\left(x_{S}, y_{S}\right)$ are the coordinates of the centroid $S$ of the triangle $A B C$. From this observation, it follows similarly to the above that for any triangle $A B C$ inscribed in $k$, the sum $f(P)=P A^{2}+P B^{2}+P C^{2}$ is minimal at the point $P$ where the ray $O S$ intersects $k$—i.e., the point on $k$ closest to $S$—and maximal at the opposite point. $S$ coincides with $O$ if and only if the triangle $A B C$ is equilateral, in which case $f(P)$ is constant for any point $P$ on $k$ (as is the case for any regular polygon with an even number of sides).

The line $O S$ is known as the Euler line of the triangle. It is well-known that, if it exists, it also passes through the orthocenter of the triangle, as well as the center of the circumcircle of the medial triangle and the pedal triangle of the triangle (the Feuerbach circle). The above result provides another property of the Euler line.

2. The value of the expression $f(P)$ is proportional to the moment of inertia of the system of mass points $A, B, C$ about an axis perpendicular to the plane of $k$ and passing through $P$, if we place equal masses at each of $A, B, C$. Our result is consistent with the mechanical theorem of Steiner: if the moment of inertia of a system of total mass $M$ about an axis $t$ passing through the center of mass $S$ of the system is $K_{0}$, then the moment of inertia about a parallel axis $t^{\prime}$ at a distance $d$ from $t$ is $K=K_{0}+M d^{2}$, so $K$ is minimal about $t$, and $d$ is minimal when $t^{\prime}$ is taken through the point on $k$ closest to $S$.

Several papers have correctly pointed out this connection. However, it is a significant detour to attempt to answer the question via this route, as the theorem cited itself is the result of mathematical reasoning.