The Pitot theorem in

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

states that in a

tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex polygon, convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This cir ...

the two pairs of opposite sides have the same total length. It is named after French engineer

Henri Pitot Henri Pitot (; May 3, 1695 – December 27, 1771) was a French hydraulic engineer and the inventor of the pitot tube. The incoming fluid in the internal tube may be blocked off where a pressure gauge can indicate the pressure, or fed to a clo ...

Statement and converse

A tangential quadrilateral is usually defined as a

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

for which all four sides are

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

to the same

inscribed circle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incente ...

. Pitot's theorem states that, for these quadrilaterals, the two sums of lengths of opposite sides are the same. Both sums of lengths equal the

semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...

of the quadrilateral. The

converse implication In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposition '' ...

is also true: whenever a convex quadrilateral has pairs of opposite sides with the same sums of lengths, it has an inscribed circle. Therefore, this is an exact characterization: the tangential quadrilaterals are exactly the quadrilaterals with equal sums of opposite side lengths.. See in particular pp. 65–66.

Proof idea

One way to prove the Pitot's theorem is to divide the sides of any given tangential quadrilateral at the points where its inscribed circle touches each side. This divides the four sides into eight segments, between a vertex of the quadrilateral and a point of tangency with the circle. Any two of these segments that meet at the same vertex have the same length, forming a pair of equal-length segments. Any two opposite sides have one segment from each of these pairs. Therefore, the four segments in two opposite sides have the same lengths, and the same sum of lengths, as the four segments in the other two opposite sides.

History

Henri Pitot proved his theorem in 1725, whereas the converse was proved by the Swiss mathematician

Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...

in 1846.

Generalization

Pitot's theorem generalizes to tangential

2n

-gons, in which case the two sums of ''alternate'' sides are equal. The same proof idea applies.{{citation , last = de Villiers , first = Michael , url = https://www.tandfonline.com/doi/abs/10.1080/0020739930240204 , title = A unifying generalization of Turnbull's theorem , journal = International Journal of Mathematical Education in Science and Technology , volume = 24 , year = 1993 , issue = 2 , pages = 65–82 , doi = 10.1080/0020739930240204 , mr = 2877281 .

References

External links

Alexander Bogomolny, "When A Quadrilateral Is Inscriptible?" at Cut-the-knot
Theorems about quadrilaterals and circles