19. Find the minimum value of
$$
\sqrt{x^{2}+(20-y)^{2}}+\sqrt{y^{2}+(21-z)^{2}}+\sqrt{z^{2}+(20-w)^{2}}+\sqrt{w^{2}+(21-x)^{2}} .
$$
19. Answer: 58 .
As shown below,
$$
\begin{aligned}
& \sqrt{x^{2}+(20-y)^{2}}+\sqrt{y^{2}+(21-z)^{2}}+\sqrt{z^{2}+(20-w)^{2}}+\sqrt{w^{2}+(21-x)^{2}} \\
= & A B+B C+C D+D A \\
= & A^{\prime} B+B C+C \bar{D}+\bar{D} \bar{A}^{\prime \prime} \\
\leq & A^{\prime} \bar{A}^{\prime \prime}=\sqrt{42^{2}+40^{2}}=58 .
\end{aligned}
$$