GeoProof — Pitot's Theorem explorer

AB + CD = BC + DA

Drag the four touch points around the circle. Every side stays tangent, so the opposite-side sums always match.

Pitot's Theorem

A tangential quadrilateral is one that has an inscribed circle touching all four sides. For any such quadrilateral ABCD, the two pairs of opposite sides have equal sums:

AB + CD = BC + DA

In this tool the quadrilateral is built directly from four tangent lines to a fixed circle, so it is always tangential — drag the touch points and the equality never breaks.

Why it works: equal tangents

From any point outside a circle, the two tangent segments to the circle are equal. So from each corner the two tangents to its neighbouring touch points match — call them a, b, c, d at A, B, C, D. Then AB = a + b, BC = b + c, CD = c + d, DA = d + a, and both sums equal a + b + c + d. Turn on "Tangent segments" to see the equal pairs.

The converse

The converse is also true (Pitot's full statement): if a convex quadrilateral has AB + CD = BC + DA, then a circle can be inscribed in it. So the equation is exactly the condition for an incircle to exist.

Familiar special cases

Every rhombus and every square is tangential (all sides equal, so both sums match). A general rectangle is not, unless it's a square.

A bit of history

It is named for Henri Pitot, the French engineer (also known for the Pitot tube), who stated it in 1725.