Recently, in connection with some work on Diophantine approximations, I encountered the following problem on triangles.

Let T be a triangle with vertices A, B, C which are, respectively, inner points of the sides a, b, c of a second triangle t. Is it always possible to move T into a new position where its vertices are inner points of t?

I give here an affirmative answer to the problem and prove, moreover, that it suffices to apply to T an arbitrarily small rotation about a suitably chosen point of the plane. I am indebted to my Manchester colleagues for a number of simplifications of this solution, arrived at when discussing the problem with them.

The purpose of this article is to describe some apparatus which is easy to construct from materials that are reasonably cheap, and which has been found useful as a visual aid in many branches of elementary mathematics. It is thought that this apparatus will satisfy the need to which attention is drawn in § 4.8 of the Trigonometry Report, although it was being used before the report was published. After the apparatus has been described, some of its applications will be briefly indicated and a more detailed account of one application will follow.

The purpose of this article is to apply a method, previously discussed in some detail, to the solving of standard elementary problems in the Calculus of Variations and to show how rapidly the solutions can be derived by introducing (p, r) and certain other coordinates. In each case a Lagrange undetermined multiplier is introduced in the course of the solution, so that the problem if not already an isoperimetrical problem is converted to one.

Let ABC be a plane triangle with D, E, F the midpoints of the sides BC, CA, AB respectively. Let L, L’ be two real intersecting straight lines in the plane. Then it can easily be seen, as in § 1 below, that if parallelograms are described on the three sides as diagonals, each with its sides parallel to L, L’, then the other diagonals of these parallelograms are concurrent at a finite point of the plane, which may be denoted by P(L, L’). An attempt to determine those points of the plane which are such points of concurrence has led to the following results.

Tonight I shall deal with some examples of mathematical reasoning which have in common only one feature: that of being surprising, or, even, of once having been surprising. We may picture them as monsters, waiting to leap on our loose ideas about mathematics and to tear them to shreds.

I shall not concern ourselves with elementary errors. Every one of you, no doubt, can “prove” that all triangles are isosceles, and that 2 = 1 by simple algebra.

The mathematics teacher has from time to time to make a decision as to how much he should say to a class about a topic which is too difficult to discuss with full mathematical rigour. This situation may arise with students who are too young or too immature in their mathematical development to appreciate a rigorous discussion, or with older students who are interested in mathematics purely as a tool which they require in the study of other subjects such as engineering and have not time to spare even if they would be interested in a rigorous development of the particular topic.