Pitot's Theorem

Pitot's Theorem for a tangential/circumscribed quadrilateral

Pitot's theorem
If a quadrilateral ABCD is tangential to or circumscribed around a circle, then the two sums of opposite¹ sides are equal, i.e. AB + CD = BC + DA.

This theorem is named after a French engineer Henri Pitot (1695-1771) who proved it in 1725.

Side-Angle Dual of Pitot
The side-angle dual of Pitot's theorem is the following: If a quadrilateral ABCD is inscribed in a circle (cyclic), then the two sums of the opposite¹ angles are equal, i.e. ∠A + ∠C = ∠B + ∠D (= 180° if convex; = 360° if crossed).

Note¹: It's useful when generalizing the result to higher circumscribed polygons to call sides AB and CD, and BC and DA, alternate sides instead of opposite sides. Similarly, it's useful when generalizing the dual result to higher cyclic polygons to call angles ∠A and ∠C, and ∠B and ∠D, alternate angles instead of opposite angles.

Pitot's Theorem for a tangential/circumscribed quadrilateral

Challenge
1) Can you explain why (prove that) your conjecture above is true?
2) If necessary, click on the Show Hint button in the sketch.

Converse
3) Formulate the converse. Can you prove or disprove the converse?
4) Check your answer to 3) above at Tangential Quadrilateral Converse.

Further Generalization
5) Can you generalize the result further to a tangential or circumscribed hexagon?
6) Check your answer to 3) above at Tangential/ Circumscribed Hexagon.

Application
7) Apply Pitot's theorem to prove the following interesting Side Divider Theorem for a Circumscribed Quadrilateral.

Published Papers
1) A classroom activity and guided proof of Pitot's theorem is given in the FREE DOWNLOAD of my book Rethinking Proof with Sketchpad (2012), 'Logical Discovery: Circum Quad', pp. 68-69; 171-173. Download the associated Sketchpad sketches as a zipped file from Rethinking GSP 5 Sketches.
2) Read my introductory paper to tangential quadrilaterals, aimed at talented high school learners: The Tangential or Circumscribed Quadrilateral in Learning & Teaching Mathematics, Dec 2020.

Related Activities
Also have a look at the following activities involving a tangential/circumscribed quadrilateral:
i) Concurrent Angle Bisectors
ii) Theorem of Gusić & Mladinić

Some Further Readings
For more on tangential/circumscribed quadrilaterals and their interesting properties, consult the following papers:
i) Worrall, C. (2004). A Journey with Circumscribable Quadrilaterals (PDF), Mathematics Teacher, Vol. 98, No. 3, pp. 192-199.
ii) Minculete, N. (2009). Characterizations of a Tangential Quadrilateral (PDF), Forum Geometricorum, 9: 113–118.
iii) Josefsson, M. (2011). More characterizations of tangential quadrilaterals (PDF), Forum Geometricorum, 11: 65–82.
iv) Josefsson, M. (2011). When is a Tangential Quadrilateral a Kite? (PDF), Forum Geometricorum, 11: 165–174.
v) Josefsson, M. (2014). Angle and Circle Characterizations of Tangential Quadrilaterals (PDF), Forum Geometricorum, 14: 1–13.
vi) Josefsson, M. & Dalcín, M. (2021). 100 characterizations of tangential quadrilaterals (PDF), International Journal of Geometry, Vol. 10, No. 4, 32 - 62.
vii) Humenberger, H. (2023). Unusual Cyclic and Tangential Quadrilaterals – An Overview. Mathematics in School, May, pp. 26-32.

Hard Challenge
For an online dynamic geometry sketch of an interesting, but quite challenging property of a tangential quadrilateral involving perpendicular bisectors of its sides go to this Webpage.

External Links
Tangential quadrilateral (Wikipedia)
Tangential Quadrilateral (Wolfram MathWorld)
A Problem in a Special Tangential Quadrilateral (Cut The Knot)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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Created by Michael de Villiers, 1 Sept 2020 with WebSketchpad, updated 22 March 2021; 2 Oct 2024; 7 April 2025.

Pitot's Theorem for a tangential/circumscribed quadrilateral

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Pitot's Theorem for a tangential/circumscribed quadrilateral