回復 37# Uukuokuo 的帖子
第 14 題
\(\begin{align}
& P\left( x,y,z \right) \\
& x+y+z=0 \\
& \\
& {{\overline{PA}}^{2}}+{{\overline{PB}}^{2}}+{{\overline{PC}}^{2}} \\
& =3{{\left( x-1 \right)}^{2}}+3{{y}^{2}}+2+{{\left( z-2 \right)}^{2}}+{{z}^{2}}+{{\left( z-4 \right)}^{2}} \\
& =3\left[ {{\left( x-1 \right)}^{2}}+{{y}^{2}}+{{\left( z-2 \right)}^{2}} \right]+10 \\
& \ge 3\times \frac{{{\left( x+y+z-3 \right)}^{2}}}{{{1}^{2}}+{{1}^{2}}+{{1}^{2}}}+10 \\
& =19 \\
\end{align}\)
等號成立於\(x=0,y=-1,z=1\)
