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.

The identity in question really is simple: it says, for u ≠ −1,

We describe two types of definite integral that look quite formidable, but dissolve into a much simpler form by an application of (1) in a way that seems almost magical.

Both types, or at least special cases of them, have been mathematical folklore for a long time. For example, case (10) below appears in [1, p. 262], published in 1922 (we are grateful to Donald Kershaw for showing us this example). However, they do not seem to figure in most books on calculus except possibly tucked away as an exercise The comprehensive survey [2] mentions the second type on p. 253, but only as a lemma on the way to an identity the authors call the ‘master formula’ We come back to this formula later, but only after describing a number of other more immediate applications.

Since April 2015 there has been no legal requirement in the UK to purchase an annuity with pension savings [1] while for those who reach state pension age on or after 6th April 2016 the UK Government changed the state pension deferral arrangements [2]. The latter refers to an arrangement whereby a pensioner can receive an enhanced state pension by deferring its uptake for an arbitrary number of years. These two changes raise certain questions for prospective pensioners which are worthy of some mathematical consideration.

An annuity is a guaranteed income for life in exchange for a certain sum of money: the pension pot. An alternative to the annuity since April 2015 is a ‘draw down scheme’ in which the pension pot can be used almost like an ordinary bank account and money periodically withdrawn. These two choices arise from ‘defined contribution’ pension arrangements. By contrast ‘defined benefit’ work-based (company) pensions allow no such choice and are not considered further here apart from the special case of the UK state pension. With an annuity a further option to consider, and one which predates the 2015 changes, is whether to take payments that are fixed or index-linked to inflation. Only the UK state pension offers a late retirement enhanced pension if its uptake is deferred.

Cyclotomy is concerned with the division of a circle into a given number of equal segments, amounting to the construction of a regular polygon, say a q-gon, so that we need to deliver the angle α = 2π / q, or the length cos α. The construction is by Euclidean means, which make use of only ruler and compasses. Now, from given lengths, sums and differences of lengths are easy to obtain and, with the compasses, products and quotients of lengths can be obtained from similar triangles using parallel lines; indeed even the length can be obtained by applying the intersecting chord theorem to a circle with diameter 1 + a. However, there is not much else one can do with the compasses, so that the length cos α has to come from the real roots of a sequence of quadratic equations with ‘suitable’ coefficients — the meaning of being suitable will be made clear later.

Gauss made the first significant contribution to the classical theory of cyclotomy in Article 365 of his famous Disquisitiones Arithmeticae [1] in 1801. He showed that the construction is possible if q = p is a Fermat prime, that is a prime of the form 22n + 1; see §7 for a necessary and sufficient condition for q. The only known Fermat primes are p = 3, 5, 17, 257, 65537; the cases p = 3, 5 and 17 correspond to the construction of the equilateral triangle, the regular pentagon, and the regular heptadecagon, the details for which Gauss gave.

In this paper, we collect some results on a problem which has been considered by many authors over the years, including Ed Pegg Jr., R. M. Kurchan, William Rex Marshall, David Neary, Richard Korf and Eric Huang. The results have been rediscovered many times by many people, including the present authors, but very little has been published, except in a little known journal which no longer appears. Some of the cases have been solved by hand while others have been covered by Korf [1, 2], using a computer. We see our main role as that of collectors of information and presenters of results, in the hope that others will be inspired to study what is an exciting and very interesting area of combinatoric geometry; we believe, however, that some of our observations, variations and upper bounds are new.

Dedicated to the memory of Russell Towle, a remarkable man who contributed so much to geometry and to other aspects of the quality of life.

We introduce an expanded notation where r + s = n, for the binomial coefficient , and then use this expanded notation to develop theorems involving 8 binomial coefficients, analogous to the Star of David Theorem, which. in its original form, involved the 6 neighbours of a given binomial coefficient in the Pascal Triangle (see Section 3), that appeared in [1,2,3,4,5,6,7,8,9].

This article is motivated by, and is meant as a supplement to, the recent paper [1]. That paper proves three geometric characterisations of triangles whose sides are in arithmetic progression, or equivalently triangles in which one of the sides is the arithmetic mean of the other two. More precisely, it gives three geometric contexts in which such triangles appear. In this Article, we supply references for the results in [1] and we provide more proofs of these results. We also add more contexts in which such triangles appear, and we raise related issues for future work. We hope that this will be a source of problems for training for, and for including in, mathematical competitions.