Question:
For a complex number $z,$ find the minimum value of
\[|z - 3|^2 + |z - 5 + 2i|^2 + |z - 1 + i|^2.\]

Answer:
Let $z = x + yi,$ where $x$ and $y$ are real numbers.  Then
\begin{align*}
|z - 3|^2 + |z - 5 + 2i|^2 + |z - 1 + i|^2 &= |x + yi - 3|^2 + |x + yi - 5 + 2i|^2 + |x + yi - 1 + i|^2 \\
&= |(x - 3) + yi|^2 + |(x - 5) + (y + 2)i|^2 + |(x - 1) + (y + 1)i|^2 \\
&= (x - 3)^2 + y^2 + (x - 5)^2 + (y + 2)^2 + (x - 1)^2 + (y + 1)^2 \\
&= 3x^2 - 18x + 3y^2 + 6y + 40 \\
&= 3(x - 3)^2 + 3(y + 1)^2 + 10 \\
&\ge 10.
\end{align*}Equality occurs when $x = 3$ and $y = -1,$ so the minimum value is $\boxed{10}.$