Tang quad — Tangential quadrilateral: opposite-side sums equal · Principia

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

P \in AB,\qquad Q \in BC,\qquad R \in CD,\qquad S \in DA.

Then the two pairs of opposite sides have equal sums:

AB + CD \;=\; BC + AD.

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

$Tangential quadrilateral ABCD with inscribed circle \odot I, tangent points P, Q, R, S on the four sides; AB + CD = BC + AD$

Opposite-side sums of a tangential quadrilateral are equal

Statement

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