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Extremal area ellipses of a convex quadrilateral

Published online by Cambridge University Press:  03 February 2017

John R. Silvester
John R. Silvester*
Affiliation:
Department of Mathematics, King's College, Strand, London WC2R 2LS e-mail: jrs@kcl.ac.uk
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Extract

In this paper we are going to generalise some standard results about the ellipses of least or greatest area, respectively circumscribing or inscribed in a triangle, to the corresponding ellipses for a convex quadrilateral.

Given ΔABC let D, E, F be the midpoints of BC, CA, AB respectively; then the medians AD, BE, CF concur at the centroid G of ΔABC. There is an ellipse with centre G, touching the sides of ΔABC at their respective midpoints. It is known as the midpoint ellipse or Steiner inellipse of ΔABC, and it is the ellipse of greatest area inscribed in ΔABC. Since AG : GD = 2 : 1, and similarly for the other medians, a central dilation (an enlargement) with centre G and scale factor –2 (or, equivalently, 2) applied to the Steiner inellipse produces a similar ellipse, also with centre G, and passing through A, B and C. This is the Steiner circumellipse of ΔABC, and it is the ellipse of least area circumscribing ΔABC. See Figure 1. For more detail, see [1, 2] and the references given there.

Type
Articles
Information
The Mathematical Gazette , Volume 101 , Issue 550 , March 2017 , pp. 11 - 26
Copyright
Copyright © Mathematical Association 2017 

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References

1. Weisstein, Eric W., ‘Steiner circumellipse’ Wolfram MathWorld, accessed October 2016 at http://mathworld.wolfram.com/SteinerCircumellipse.html Google Scholar
2. Weisstein, Eric W., ‘Steiner inellipse’ Wolfram MathWorld, accessed October 2016 at http://mathworld.wolfram.com/SteinerInellipse.html Google Scholar
3. Horwitz, Alan, Ellipses of minimal area and of minimal eccentricity circumscribed about a convex quadrilateral, Aust. J. Math. Anal. App., 7 (2010) Article 8, also available at: http://arXiv.org/abs/0707.2092v1 Google Scholar
4. Hardy, G. H., Pure mathematics (10th edn.), Cambridge University Press (1955), p. 497, Ex. 9.Google Scholar
5. Semple, J. G. and Kneebone, G. T., Algebraic projective geometry, Oxford University Press (1952).Google Scholar