1. A triangle $A B C$ is to be constructed so that $A$ is at $(3,2), B$ is on the line $y=x$, and $C$ is on the $x$-axis. Find the minimum possible perimeter of $\triangle A B C$.
Answer: $\sqrt{26}$
Solution: Let $D(2,3)$ be the reflection of $A$ with respect to $y=x$, and $E(3,-2)$ the refection of $A$ with respect to the $x$-axis. Then if $B$ is on $y=x$ and $C$ is on the $x$-axis, then $|A B|=|D B|$ and $|A C|=|C E|$. Then the perimeter of $\triangle A B C$ is
$$
|A B|+|B C|+|A C|=|D B|+|B C|+|C E| \geq|D E|=\sqrt{26} .
$$

Clearly, this lower bound can be attained.