The work presented here was prompted by curiosity concerning a structural detail of the ironwork supporting the overhead cover of the railway platform at Stockport, Cheshire. The detail – see the front cover which shows the roof support – comprises an elliptical ‘quadrant’ joined to vertical and horizontal members of the supporting structure, with a circular ring fitted interstitially so as to touch the elliptical arc and the two straight members. The question arises: what is the radius of the circular ring in terms of the parameters of the ellipse?

This article presents interesting generalisations of three well-known results related to pedal triangles and distances, and the Simson line.

The triangle whose vertices are the feet of the perpendiculars from a point P inside a triangle ABC to each of its sides AB, BC and AC, is called a pedal triangle.

The defining property of a rectangle is that it is a quadrilateral whose angles are all equal (or are right angles, which amounts to the same thing).

The opposite sides of a rectangle are equal. If a and b are the lengths of adjacent sides of a rectangle, such a rectangle can be described as an abab rectangle (or an (ab)2 rectangle). The area of the rectangle is ab.

As far as I understand the latest version of the national curriculum and the new A level syllabuses, which make no serious reference to irrational numbers, students leaving school will not necessarily know that there are two sorts of rectangle: those coverable by a set of identical squares, which can therefore be drawn on squared paper; and those not, such as the golden rectangle. An (ab)2 rectangle is coverable by squares if, and only if, a/b is rational.

Conies are often presented through the following definition: The conic Γ with focus F, directrix l and eccentricity e is the set of points Y of the Euclidean plane E2 such that d(Y, F) = e d(Y, l). Of course, the main interest in this focus-directrix property is to allow a unified approach to the three conies. Surprisingly, a simple construction of Γ given F, l, e seldom follows. The aim of this note is to fill the gap by describing … just one construction for the three conies! Readers are strongly encouraged to try out this construction, either by hand, using a ruler and a set-square, or by using a computer package. As is to be expected, the tangents to Γ are also easily obtained. Moreover, the transformation underlying the whole process leads to some properties of the conies in a simple and elegant way. A few examples will be given.

Fermat (1601-1665) is well-known for offering mathematical results without stating their proofs. In Mahoney’s fine mathematical biography, suggestions are made giving possible lines of reasoning which Fermat may have used, suggestions which are easily recognised by those familiar with number theory. This article offers some conjectured reconstructions of Fermat’s reasoning which may be more accessible to a beginner since they are linked to pattern recognition, and capitalise on the special cases with which Fermat illustrated his ideas. Generic examples played an essential part in Fermat’s exposition and may well have played a larger part in his proofs than would be respectable in a textbook nowadays.

The nth triangular number is defined to be the sum of the first n positive integers:

Thus

In a letter to Mersenne in 1638 [1, p. 61], Fermat claimed that every positive integer can be written as a sum of at most three triangular numbers. This remarkable result was eventually proved by Gauss in 1796, at the age of 19.

A rational number is said to be expressed in Egyptian form (or Egyptian presentation) if it is written as a sum of fractions where each fraction has the numerator 1 and all of the denominators are distinct. The individual terms of die sum are often referred to as unit fractions. The problem of finding such expressions, and many related questions, has a long history, apparently beginning with Ahmes in the Rhind Papyrus. It is for this reason that we talk of ‘Egyptian’ form.

Games arouse enthusiasm and present challenge. Mathematical games add a form of scientific discovery. The beauty of mathematical games lies in the challenge of discovering patterns hidden in a (sometimes very simple) set of playing rules. Once a pattern is identified, the triumph of discovery brings the satisfaction of mental achievement and the joy of defeating an opponent. When the games can be programmed and colourfuUy demonstrated on a computer, the sense of satisfaction can expand even further.