Opposite-side sums of a tangential quadrilateral are equal

Depends on: Two tangents from an external point have equal length (Two tangents from an external point have equal length).

Statement

Let convex quadrilateral $ABCD$ be circumscribed about a circle $\odot I$ — i.e. there is an inscribed circle $\odot I$ tangent to all four sides $AB$, $BC$, $CD$, $DA$. Let the points of tangency be

Then the two pairs of opposite sides have equal sums:

This is the opposite-side-sum theorem for a tangential quadrilateral (also called Pitot's theorem).