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Conics and convexity

Published online by Cambridge University Press:  23 January 2015

K. Robin McLean*
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL

Extract

In [1], W. D. Munn proved the following result.

Theorem 1: Infinitely many ellipses pass through the four vertices of a given convex quadrilateral.

Much of the geometry that I studied as an undergraduate in the 1950s concerned complex projective space, in which convexity plays no part. So I found Theorem 1 especially piquant and sought to understand it better. This article is the result. After examining the convexity of quadrilaterals in general, especially those inscribed in conics, I consider the following problem. Let P be a variable point in the plane, distinct from the vertices of a given convex quadrilateral ABCD. It is well known that there is a unique conic, S (P), through the five points A, B, C, D and P. How does the nature of this conic depend on the position of P? As a spin-off, we get a very short proof of Theorem 1. Finally I look at what happens when the quadrilateral ABCD is not convex. In this case, S (P) is always a hyperbola, but the distribution of A, B, C and D on its branches is still of interest.

Type
Articles
Copyright
Copyright © Mathematical Association 2014 

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References

1. Munn, W. D., Ellipses circumscribing convex quadrilaterals, Math. Gaz. 92 (November 2008) pp. 566568.Google Scholar
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