Given the triangle $A B C$ and the line $e$. For which point $P$ on $e$ will the distance $P A^{2}+2 P B^{2}+3 P C^{2}$ be the smallest?
I. solution. By orthogonally projecting points $A, B, C$ onto line $e$, let the resulting points be $A_{1}, B_{1}, C_{1}$, respectively. According to the Pythagorean theorem,

$$
P A^{2}+2 P B^{2}+3 P C^{2}=\left(P A_{1}^{2}+2 P B_{1}^{2}+3 P C_{1}^{2}\right)+\left(A A_{1}^{2}+2 B B_{1}^{2}+3 C C_{1}^{2}\right)
$$

Identifying line $e$ with the number line, let the coordinates of points $A_{1}, B_{1}, C_{1}$ be $a, b, c$, respectively, and the unknown coordinate of point $P$ be $x$. Since $A A_{1}^{2}+2 B B_{1}^{2}+3 C C_{1}^{2}$ is constant, the sum in question is minimized when

$$
P A_{1}^{2}+2 P B_{1}^{2}+3 P C_{1}^{2}=(x-a)^{2}+2(x-b)^{2}+3(x-c)^{2}
$$

is minimized. Completing the square for this quadratic expression in $x$,

$$
6\left(x-\frac{a+2 b+3 c}{6}\right)^{2}+\left(a^{2}+2 b^{2}+3 c^{2}\right)-\frac{(a+2 b+3 c)^{2}}{6}
$$

is obtained. This is clearly minimized when

$$
x=\frac{a+2 b+3 c}{6}=\frac{\frac{a+c}{2}+b+c}{3}
$$

In this case, point $P$ is precisely the orthogonal projection of the centroid of triangle $BCD$ onto line $e$, where $D$ is the midpoint of segment $AC$.

II. solution. Let $R$ be a point such that the vectors $\overrightarrow{R A}=\mathbf{a}, \overrightarrow{R B}=\mathbf{b}$, and $\overrightarrow{R C}=\mathbf{c}$ satisfy the relation $\mathbf{a}+2 \mathbf{b}+3 \mathbf{c}=\mathbf{0}$, and let $\overrightarrow{R P}=\mathbf{p}$. We will prove later that such a point $R$ exists. Then,

$$
\begin{gathered}
P A^{2}+2 P B^{2}+3 P C^{2}=(\mathbf{a}-\mathbf{p})^{2}+2(\mathbf{b}-\mathbf{p})^{2}+3(\mathbf{c}-\mathbf{p})^{2}= \\
=\left(\mathbf{a}^{2}+2 \mathbf{b}^{2}+3 \mathbf{c}^{2}\right)+6 \mathbf{p}^{2}-2 \mathbf{p}(\mathbf{a}+2 \mathbf{b}+3 \mathbf{c})=\left(\mathbf{a}^{2}+2 \mathbf{b}^{2}+3 \mathbf{c}^{2}\right)+6 \mathbf{p}^{2}
\end{gathered}
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_37ae5e6fd78f90193e18g-1.jpg?height=458&width=724&top_left_y=1267&top_left_x=690)

Since $\mathbf{a}, \mathbf{b}$, and $\mathbf{c}$ are fixed vectors, the value of the expression is minimized when $\mathbf{p}^{2}$ is minimized. This occurs when the length of segment $RP$ is minimal, i.e., the desired point is precisely the orthogonal projection of $R$ onto line $e$.

We still need to determine the points $R$ that satisfy the relation $\mathbf{a}+2 \mathbf{b}+3 \mathbf{c}=\mathbf{0}$. We will show that there is exactly one such point, and it is the centroid of triangle $BCD$, where $D$ is the midpoint of segment $AC$.

If $\overrightarrow{R D}=\mathbf{d}$, then from the properties of the centroid, $\mathbf{b}+\mathbf{c}+\mathbf{d}=\mathbf{0}$, and since $D$ is the midpoint, $\mathbf{d}=(\mathbf{a}+\mathbf{c}) / 2$, thus

$$
\mathbf{0}=\mathbf{b}+\mathbf{c}+\mathbf{d}=\mathbf{b}+\mathbf{c}+\frac{\mathbf{a}+\mathbf{c}}{2}=\frac{\mathbf{a}+2 \mathbf{b}+3 \mathbf{c}}{2}
$$

Therefore, this point satisfies the requirement.

If for some point $T$, $\overrightarrow{T A}+2 \overrightarrow{T B}+3 \overrightarrow{T C}=\mathbf{0}$ holds, then

$$
\begin{aligned}
\mathbf{0} & =\overrightarrow{T A}+2 \overrightarrow{T B}+3 \overrightarrow{T C}=(\overrightarrow{T R}+\overrightarrow{R A})+2(\overrightarrow{T R}+\overrightarrow{R B})+3(\overrightarrow{T R}+\overrightarrow{R C})= \\
& =6 \overrightarrow{T R}+\mathbf{a}+2 \mathbf{b}+3 \mathbf{c}=6 \overrightarrow{T R}
\end{aligned}
$$

thus $\overrightarrow{T R}=\mathbf{0}$, meaning $T$ coincides with $R$.

Remark. If a triangle $ABC$ and real numbers $\alpha, \beta, \gamma$ are given such that $\alpha+\beta+\gamma \neq 0$, it can be shown that there is exactly one point $P$ for which the relation $\alpha \overrightarrow{P A}+\beta \overrightarrow{P B}+\gamma \overrightarrow{P C}=\mathbf{0}$ holds. Furthermore, for any point in the plane of the triangle, there exists a unique ordered triplet $(\alpha, \beta, \gamma)$, up to scalar multiplication, for which the relation holds. The interested reader can find a detailed description of this in Hajós Gy.: Bevezetés a geometriába (Introduction to Geometry) in the section on barycentric coordinates, 35.5.