Pitot's Theorem

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.

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 the two sums of the opposite¹ angles are equal, i.e. ∠A + ∠C = ∠B + ∠D (= 180° if convex; = 360° if crossed).

Note¹: Sometimes 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 sometimes 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

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.
3) Formulate the converse. Can you prove or disprove the converse?
4) Check your answer to 3) above at Tangential Quadrilateral Converse.
5) Can you generalize the result further to a tangential or circumscribed hexagon?
6) Check your answer to 3) above at Tangential/ Circumscribed Hexagon.
7) Apply Pitot's theorem to prove the following interesting Side Divider Theorem for a Circumscribed Quadrilateral.

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

For even more interesting properties of tangential/circumscribed quadrilaterals, consult the following papers:
i) Minculete, N. (2009). Characterizations of a Tangential Quadrilateral (PDF), Forum Geometricorum, 9: 113–118.
ii) Josefson, M. (2011). More characterizations of tangential quadrilaterals (PDF), Forum Geometricorum, 11: 65–82.
iii) Josefson, M. & Dalcín, M. (2021) & . 100 characterizations of tangential quadrilaterals (PDF), International Journal of Geometry, Vol. 10, No. 4, 32 - 62.

Read my introductory paper to tangential quadrilaterals, aimed at talented high school learners: The Tangential or Circumscribed Quadrilateral in Learning & Teaching Mathematics, Dec 2020.

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

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

Pitot's Theorem

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Pitot's Theorem