That Spring morning, the bent old man sat down at his large oak desk with his pen momentarily poised over the familiar blue foolscap sheet. He furtively glanced around the drawing room with its old china and flowers and books in abundance and wrote in one easy movement: The principles of book-keeping by double entry. He looked up once more but this time allowed a minute to gaze out through the window to the Cam gently meandering through the meadows towards the Cambridge colleges.

This article considers the use of a range of computer tools for aiding geometric exploration, together with a suitable algebraic representation for objects connected with the triangle. The techniques are applied to produce new results in the geometry of the triangle. This is an extended version of part of the contribution: How do computers change the way we do mathematics? given at the Association’s 1994 Easter Conference.

In the geometry of the triangle, the Feuerbach quadrangle comprises the four points of contact between the nine point circle and the four tritangent circles. By omitting each of the four points in turn, we create four Feuerbach triangles. When the basic triangle is isosceles we find that two of the four points of contact coincide and the resultant single Feuerbach triangle is also isosceles. In the spirit of Steiner-Lehmus we address the converse question – if one of the Feuerbach triangles is isosceles, is the basic triangle necessarily isosceles? The answer is negative.

Consider the statement (whose truth is obvious):

This is known as a pigeon-hole principle (or Dirichlet’s). It is one of the strategies which is often used for problem-solving. Although it is one of the most powerful tools of combinatorics, the principle can be used successfully in other branches of mathematics. Below you can find a collection of problems which can be solved easily by applying this principle to geometrical objects.

Over the years 1830-1832, there appeared three volumes of a mathematical textbook in Welsh Elfenau Rhifyddiaeth (the elements of arithmetic) by John William Thomas. The book has long been forgotten and apparently only one specimen now remains, in the Salesbury Library in the Cardiff College of the University of Wales. The treatment and subject matter of the work are in places surprising for that era, and it is worth bringing the contents of the book to light.

For a long time prime numbers have attracted the attention of mathematicians, especially those primes that possess some sort of a symmetry. The mysterious repunits An = 111… 1(n), whose decimal representations contain only units, form an important class of them. For a repunit to be prime the number n of its digits must be also prime. But this condition is far from being sufficient: for instance, A3 = 111 = 3.37 and A5 = 11111 = 41.271. Some of the repunits are nonetheless prime: A2, A19, A23, A317 and A1031, are the only known examples. The question of primeness of the repunits was discussed by M. Gardner and later in [2–4]. It is not clear whether the number of prime repunits is finite or infinite.

With so much changing almost daily in schools and education, it has been difficult to decide on a title for this address and even more difficult to finalize a text. Some of the titles I’ve considered are shown here.