Theorem

Pitot's theorem for a tangential quadrilateral

Geometry • Quadrilateral • Tangential

Formula

A B + C D = B C + D A

If a convex quadrilateral

A B C D

has an inscribed circle touching all four sides, then

A B + C D = B C + D A

; the converse also holds, so a convex quadrilateral with equal opposite-side sums admits an incircle.

How to use it

1Label the tangent length from each vertex $p, q, r, s$ ; the sides are then $p + q$ , $q + r$ , $r + s$ , $s + p$ and both opposite sums come to $p + q + r + s$ .
2Use it to find a fourth side when three are known and the quadrilateral is known to be tangential.
3Any four positive tangent lengths build such a quadrilateral, which is how asymmetric test cases are constructed.

Worked example

A quadrilateral with an inscribed circle has three consecutive sides

7

9

and

6

cm. Find the fourth.

Step-by-step

1
$A B + C D = B C + D A \Rightarrow 7 + 6 = 9 + D A$
Why: The sides $7$ and $6$ are opposite, as are $9$ and the unknown.
2
$D A = 4$ cm

Answer

4

Watch out

It concerns opposite sides, not adjacent ones — a square satisfies both readings, so a square is useless as a test case and an asymmetric example is needed. The cyclic quadrilateral's corresponding result is about opposite angles summing to

18 0^{\circ}

, an entirely different statement.

Builds on

The two tangents from an external point

Also known as

Pitot theorem, tangential quadrilateral, inscribed circle in a quadrilateral, circumscribed quadrilateral

Pitot's theorem for a tangential quadrilateral

Formula

Worked example

Appears in past papers

2019 (II)

2016 (II)