Pitot theorem on:
[Wikipedia]
[Google]
[Amazon]
In
Alexander Bogomolny, "When A Quadrilateral Is Inscriptible?" at Cut-the-knot
Theorems about quadrilaterals and circles
In geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the Pitot theorem, named after the French engineer Henri Pitot
Henri Pitot (; May 3, 1695 – December 27, 1771) was a French hydraulic engineer and the inventor of the pitot tube.
In a pitot tube, the height of the fluid column is proportional to the square of the velocity of the fluid at the depth of the ...
, states that in a 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 ...
(i.e. one in which a circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
can be inscribed) 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 na ...
of the quadrilateral.
The theorem is a logical consequence of the fact that two tangent line segments from a point outside the circle to the circle have equal lengths. There are four equal pairs of tangent segments, and both sums of two sides can be decomposed into sums of these four tangent segment lengths. 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 proposi ...
is also true: a circle can be inscribed into every convex quadrilateral in which the lengths of opposite sides sum to the same value.. See in particular pp. 65–66.
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.
Pitot's theorem generalizes to tangential 2''n''-gons, in which case the two sums of ''alternate'' sides are equal.{{citation, first=Michael, last=1de Villiers, 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.
See also
* Tangent quadrilateral theorem and inscribed circlesReferences
External links
Alexander Bogomolny, "When A Quadrilateral Is Inscriptible?" at Cut-the-knot
Theorems about quadrilaterals and circles