Set of pitot

${\displaystyle {\begin{aligned}&{\overline {AB}}+{\overline {CD}}\\=&(a+b)+(c+d)\\=&(b+c)+(a+d)\\=&{\overline {BC}}+{\overline {DA}}\end{aligned}}}$

The Pitot theorem , named after the French engineer Henri Pitot is a statement in elementary geometry and describes a property of tangent squares . This means that in a tangent quadrilateral the two sums of the lengths of opposite sides are equal.

In a tangent quadrilateral ABCD the following applies:

${\displaystyle {\overline {AB}}+{\overline {CD}}={\overline {BC}}+{\overline {DA}}.}$

This equality arises directly from the symmetry property of the circle, since due to this the two tangent sections starting from a point are of equal length (see drawing).

The converse of the theorem also applies, that is, if the above equation is satisfied in a convex quadrilateral ABCD , then it is also a tangent quadrilateral . Pitot's theorem and its inverse are collectively referred to as the tangent quadrilateral theorem.

Henri Pitot proved his theorem in 1725 and the reverse was proven in 1846 by the Swiss mathematician Jakob Steiner .

literature

• Martin Josefson: More characterizations of Tangential Quadrilaterals . Forum Geometricorum, Volume 11, 2011, pp. 65–82, especially pp. 65–66
• Siegfried Krauter, Christine Bescherer: The Elementary Geometry Experience: A workbook for independent and active discovery . Springer, 2012, ISBN 9783827430250 , pp. 77-78
• Lorenz Halbeisen, Norbert Hungerbühler, Juan Läuchli: With harmonious proportions to conic sections: pearls of classical geometry . Springer 2016, ISBN 9783662530344 , p. 21 ( excerpt )

Web links

• Alexander Bogomolny, "When A Quadrilateral Is Inscriptible?" at cut-the-knot