Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-04T21:33:03.827Z Has data issue: false hasContentIssue false

New characterisations of bicentric quadrilaterals

Published online by Cambridge University Press:  12 October 2022

Martin Josefsson*
Affiliation:
Västergatan 25d, 285 37 Markaryd, Sweden e-mail: [email protected]

Extract

A bicentric quadrilateral is a convex quadrilateral that can have both an incircle (it is tangential) and a circumcircle (it is cyclic), see Figure 1. We know of only a dozen characterisations of bicentric quadrilaterals published before. In all of them the starting point is either a tangential or a cyclic quadrilateral, which then must satisfy some condition in order also to be of the other type. Before we proceed to prove seven new such necessary and sufficient conditions for bicentric quadrilaterals, we review one characterisation and one property of tangential quadrilaterals that we will apply later.

Type
Articles
Copyright
© The Authors, 2022. Published by Cambridge University Press on behalf of The Mathematical Association

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.Google Scholar
Durell, C. V. and Robson, A., Advanced trigonometry, Dover (2003).Google Scholar
Josefsson, M., More characterizations of cyclic quadrilaterals, Int. J. Geom. 8(2) (2019) pp. 1432.Google Scholar
Josefsson, M., On the inradius of a tangential quadrilateral, Forum Geom. 10 (2010) pp. 2734.Google Scholar
Josefsson, M., The area of a bicentric quadrilateral, Forum Geom. 11 (2011) pp. 155164.Google Scholar
Covas, M. A., Problem 2209, Crux Math. 24(2) (1998) pp. 112113.Google Scholar
Josefsson, M., Characterizations of bicentric quadrilaterals, Forum Geom. 10 (2010) pp. 165173.Google Scholar
2004 All-Russian Olympiad, AoPS Online, available at https://artofproblemsolving.com/community/c5164 Google Scholar
Quadrilateral which is a cyclic quadrilateral and tangent qu, AoPS Online, accessed March 2022 at https://artofproblemsolving.com/community/c6h5510 Google Scholar