Putnam Perimeter!

Consider the triangle with points $A(p,q),B(a,0),C(b,b)$ where $A$ is fixed. Reflect $A$ across $x$ -axis to get $A^\prime (p,-q)$ and across $x=y$ to get $A^*(q,p)$ . So the perimeter is $AB+BC+CA=A^\prime B+BC+CA^*$ . Now it's clear that the minimum is achieved when $A^\prime BCA^*$ is a straight line.

So the minimum perimeter is the length of $A^\prime A^*$ , that is $\boxed{\sqrt{2\left(p^2+q^2\right)}.}$

Some additional calculations also reveal that the minimum perimeter triangle is a right triangle.