 Taxi distance convolution on a rectangular road cycle

For a taxi starting at any vertex of a rectangle with side lengths $s,t$, a uniform pickup vertex gives approach-distance law uniform on $0,s,t,s+t$. Conditional on that pickup, a uniform distinct destination gives passenger-distance law uniform on $s,t,s+t$, independent of the pickup vertex. Hence the two distance variables are independent despite sharing that vertex. Their convolution has twelve equally likely pairs. Its mean is $7(s+t)/6$ and its variance is $(s^2+t^2)/4+2(s^2+t^2-st)/9$. Repeated numerical sums combine their multiplicities when some distances coincide.