Bob,

If the inscribed circle with center O touches ABCD at W (on AB), X(on BC), Y(on CD) and Z (on DA), then triangles OAW, OAZ are congruent (why?) as are three other pairs. [Draw and label the diagram]

So AW = AZ [and three other equalities - find them!]

Deduce from this that AB + CD = BC + AD for any quadrilateral with an inscribed circle. (Less obviously, this is if-and-only-if.)

Solve your problem.

Good hunting!
RD