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Geometry is a very interesting, applicable and beautiful part of mathematics. However, geometry is often difficult for students to understand and demanding for teachers to teach [1]. Constructing proofs in geometric problems turns out to be particularly difficult, even for high attaining students [2]. Sometimes, students do not even know where to start when trying to solve these [3].

A pleasurable aspect of mathematics and its teaching is to review the diversity of ways in which theorems are proved. Especially in elementary branches, there are various kinds of proof: using (or avoiding) spatial geometry, analytic or coordinate geometry, common algebra, vectors, abstract algebras, matrices, determinants, the differential and integral calculus, and maybe mixtures thereof. Further, sometimes a proof of one kind is elegant while another is clumsy, or one proof of a theorem suggests why it follows while another proof is not perspicuous. There is also the question of whether a proof is direct or indirect (for example, proofby contradiction).

Mathematical students are usually familiar with the parametric coordinates (at2 , 2at) of a point on a parabola, and with the eccentric-angle coordinates of a point on an ellipse, and with similar coordinates for a hyperbola. Such coordinates are probably welcomed by the student just because they reduce by one the number of letters and the number of equations with which he has to deal. If at2 is substituted for x1 , and 2at for y1 , but everything else is done in just the same way as before, parametric methods are not really being used. Parametric equations are something more than convenient coordinates.

It is a far cry from the Conics of Apollonius to the theory of linear spaces and matrix algebra and many men have helped to build the road between. This article is about one of the builders whose work for a time was discarded and whose name was almost forgotten. But his name finds frequent mention now, when the latter aspect of geometry bears so little resemblance to the former, and metrical geometry has given place to projective geometry with its non-metrical homogeneous coordinates, which make it possible to publish a book on coordinate geometry which does not contain a single figure.

The present paper is a sequel to two others which have appeared in the Gazette (July, 1941 and December, 1957) and in which the possibility of attacking a number of familiar triangle theorems by cartesian methods was successfully investigated. The same method is here applied to Morley’s and Pascal’s Theorems.

The question of sign constitutes a real difficulty to the intelligent boy at the outset of his study of Coordinate Geometry. At the beginning of his Trigonometry he is told OP must be considered always positive, but later on he will find some authorities giving a point in the third quadrant as ( - r, θ), while others prefer (r, θ + π). The perpendicular distance of (h, k) from ax + by + c = 0 is given by ± (ah + bk +c)/(a2 + b2)½, and sign seems to matter, but usually the pupil is told that he only wants to know how far off (h, k) is, and he is advised to stick to the absolute value. But a little later on he wants the equations of the bisectors of the angles between two given lines, and then he is blamed for not remembering that signs matter a good deal.

Consider the hyperbola y = 1/x. (It will be convenient to restrict attention to that part for which x > 1.) Everyone will agree that the x-axis is an “asymptote” to this curve; however, different people are likely to advance different reasons for this fact.

Plane Analytical Geometry, and elementary Trigonometry, are the beginnings of the systematic application of mathematical analysis to Geometry, i.e. of systematic expression in terms of “number”—which is essentially “real” number.Complete application of the generality of the algebra (within the restriction to real number) is, of course, the power of the method; and, in particular, sign of the (real) numbers is significant throughout.

When it is said that Descartes was the inventor of analytical geometry, the impression is often given that by Cartesian Geometry is simply meant a system in which the position of a point is determined by means of its coordinates referred to two fixed straight lines, either at right angles or oblique. A little consideration shows us that it is not correct to say that this system was invented by Descartes, as we find that Apollonius, Bk. V, Prop. 52, gives the values of the coordinates of the centre of curvature of a point on a conic referred to two rectangular axes. It will no doubt interest the readers of the Mathematical Gazette to know what is was that Descartes invented, and what led him to it.

The strength of analytical methods is generality. It is not quite easy to bring out effectively the essential generality of elementary Analytical Geometry, In that connection the following points appear to be of fundamental importance:1. (i) If P, Q, R are collinear points, the ratio PR : PQ is positive or negative, according as the directions * PQ, PR are the same or opposite.(ii) (1) Cartesian coordinates of the representative point P of a given plane are determined by reference to chosen axial lines xOx′, yOy′ in the plane, and the question of sign may be put thus, using (i) : If A, B are the points at unit distance from O on the half-lines Ox, Oy, respectively, and if the parallels through P to the axial lines intersect the x-axis at M and the y-axis at N, the coordinates x, y of P are given by x = OM: OA, y = ON: OB. They are essentially one-valued functions of the position of P.