The idea of the representation of a linear system of plane curves by means of a flat space is of fundamental importance in algebraic geometry. In this paper some details of the representation of circles are worked out, and an attempt is made to give a three-dimensional picture of certain parts of circle geometry The theorems given in italics are thought to be new, although in a subject with such an enormous literature it might almost be said that no theorem can be a new one.

The reader should first try the following exercise, which motivates the later work.

Exercise: Prove that the set of real matrices

forms a group under the usual matrix multiplication.

This exercise comes with a guarantee that it will provoke at least a doubletake when first encountered. The usual reaction is to protest that these matrices are obviously singular, or to complain that the identity is missing from the set. Of course those familiar with this example can feel suitably smug in their rapid dismissal of these criticisms, but what was your reaction the first time you saw it?

In studying an axiomatic subject, such as Group Theory, it is desirable to use the smallest set of axioms possible. In this short article we give three examples in which sets of classical axioms can be reduced.

For any three points X, Y, Z, let ⊙XYZ denote the circle through X, Y, Z (the circumcircle of ∆XYZ) or, if X, Y, Z happen to be collinear, the line XYZ. (We shall often regard lines as special circles, circles of infinite radius.) This paper is about the following theorem, and extensions of it:

Theorem 1: Given ∆ABC and a point P, reflect ⊙PBC, ⊙APC, ⊙ABP in the lines BC, AC, AB respectively. Then the three reflected circles have a common point, Q (see Figure 1).

I do not know if this theorem is new, but I have not come across it in the literature. The reader is invited to prove it by angle-chasing, using circle theorems: let two of the reflected circles meet at Q and then prove that this point lies on the third reflected circle. This method is rather diagram-dependent, and does not seem to lead to the extensions of Theorem 1 referred to above in any very obvious manner, so we shall adopt a different approach.

It was conjectured by Fermat that if n > 2, the equation has no solution in positive integers. If n is composite, say n = lm, the equation implies Am + Bm = Cm, where A = al, etc. So the insolubility of the equation with exponent m ensures the insolubility of the equation with exponent n.

Heron triangles are those with integer sides and integer area. It is well known how to construct them as the union or difference of a pair of integersided right-angled triangles with a common side. For example, the triangles with sides 8, 15, 17 and 20, 15, 25 may be united, with common side 15, to form an acute Heron triangle with altitude 15, base 28 and other two sides 17 and 25. Its area is 210. Alternatively, their difference may be formed to create an obtuse Heron triangle with altitude 15, base 12 and other two sides 17 and 25. Its area is 90. Triangles with rational sides and rational area may be enlarged to have integer sides and integer area, and so may be classed as Heron triangles also. There have been many articles about Heron triangles in recent times, both in the Gazette [1,2] and elsewhere [3,4, 5], to mention just a few. This is not surprising as the number theory involved has a direct and pleasing application.

Geometrically considered the sateen arrangement is a certain regular method of arranging points on squared paper, each point being placed at the centre of a square. In textile work the sateen arrangement is used, (1) as a basis for actual weaves, (2) as a mode of distributing over the surface of a cloth the repeats of an ornamental design.

I was intrigued by Brian Bolt’s note “The Home Stretch”, as it had not occurred to me to look at Steiner trees spanning networks in three dimensions before. Nor did I find any references to them in the (admittedly limited) literature I consulted, though plenty has been written about the two dimensional case (see, eg, Gardner, Wells, MacKinnon). I was quickly able to improve on Bolt's tree, and decided to investigate networks for other shapes.