Concept

Pitot theorem

The Pitot theorem in geometry states that in a tangential quadrilateral the two pairs of opposite sides have the same total length. It is named after French engineer Henri Pitot. A tangential quadrilateral is usually defined as a convex quadrilateral for which all four sides are tangent to the same inscribed circle. 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 of the quadrilateral. The converse implication 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. One way to prove the Pitot 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. Henri Pitot proved his theorem in 1725, whereas the converse was proved by the Swiss mathematician Jakob Steiner in 1846. Pitot's theorem generalizes to tangential -gons, in which case the two sums of alternate sides are equal. The same proof idea applies.

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Related concepts (5)
Tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals.
Bicentric quadrilateral
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral.
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. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter s.
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