Hostname: page-component-7c8c6479df-xxrs7 Total loading time: 0 Render date: 2024-03-28T13:16:24.888Z Has data issue: false hasContentIssue false

Pitot's theorem, dynamic geometry and conics

Published online by Cambridge University Press:  17 February 2021

A. F. Beardon*
Affiliation:
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB e-mail: afb@dpmms.cam.ac.uk

Extract

It is well known that a convex quadrilateral is a cyclic quadrilateral if, and only if, the sum of each pair of opposite angles is π. This result (which gives a necessary and sufficient condition for the existence of a circle which circumscribes a given quadrilateral) is beautifully complemented by Pitot’s theorem which says that a given convex quadrilateral has an inscribed circle if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair. Henri Pitot, a French engineer, noticed the easy part of this result in 1725 (see Figure 1), and the converse was first proved by J-B Durrande in 1815. Accordingly, we shall say that a convex quadrilateral is a Pitot quadrilateral if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair.

Type
Articles
Copyright
© The Mathematical Association 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Josefsson, M., On Pitot's theorem, Math. Gaz. 103 (July 2019) pp. 333337.10.1017/mag.2019.70CrossRefGoogle Scholar
Siebeck, J., Ueber eine neue analytische Behandlungweise der Brennpunkt, J. Riene Angew. Math. 64 (1864) pp. 175182.Google Scholar
Kalman, D., An elementary proof of Marden's theorem, Amer. Math. Monthly 115 (2008) pp. 330338.10.1080/00029890.2008.11920532CrossRefGoogle Scholar
Linfield, B., On the relation of the roots and poles of a rational function to the roots of its derivative, Bull. Amer. Math. Soc. 27(1) (1920) pp. 1721.10.1090/S0002-9904-1920-03350-1CrossRefGoogle Scholar
Marden, M., Geometry of polynomials (2nd edn.), Mathematical Surveys No. 3, Amer. Math. Soc, Providence, Rhode Island (1966).Google Scholar
Minda, D. and Phelps, S., Triangles, ellipses and cubic polynomials, Amer. Math. Monthly 115 (2008) pp. 679689.CrossRefGoogle Scholar
Beardon, A. F., Creative Mathematics, Cambridge University Press (2009).Google Scholar
Rigby, J. F., Cycles and tangent rays, Math. Mag. 64 (1991) pp. 155167.10.1080/0025570X.1991.11977599CrossRefGoogle Scholar
Magnus, W., Non–Euclidean tesselations and their groups, Academic Press, New York (1974).Google Scholar