If the above phrase appeared in an A-level paper the Board responsible would receive many protests: “a contradiction in terms”, “impossible”. But is it? Our attention was drawn to this via this A-level problem from the J.M.B.

(Further Maths, Paper II, 1988)

The following, written at the reguest of the Editor of the Gazette, gives an account of the mathematical part of the remarks made by the lecturer at the Annual Meeting of the Association last January. There are added some words dealing with geometry on a sphere, in regard to which some questions were raised at the meeting.

The quadratic equation is a central topic in high school algebra. It provides the simplest generalisation of the familiar linear equation, and finding its roots introduces students to a non-trivial problem that requires the application of new techniques, such as completing the square and/or factorisation into linear factors involving the roots. It also introduces the student to the phenomenon of repeated roots, which opens the door to a discussion of multiplicities of roots. Furthermore, it naturally exposes the student to the case where the equation has no real roots, a phenomenon that could also be used to introduce the student to complex numbers.

The purpose of this article is to show how elementary properties of the cubic curve can give rise to an abstract structure with interesting properties and to investigate some of the simpler of these.

Introduction, (a) Let M, M’ be two isogonal conjugate points with respect to a triangle (T) = ABC: MP, M’P’; MQ, M’Q’; MR, M’R’ the perpendiculars from M, M’ upon the sides BC, CA, AB of (T), respectively. The triangles (M) = PQR, (M’) = P’Q’R’ are the pedal triangles of the points M, M’ for (T), and the common circumcircle (L) of those two triangles is the pedal circle of M. M’ for (T) [1, pp. 238–242].