Triangles have been used in decorative patterns from the earliest times. Some of the first geometrical discoveries were to do with properties of triangles especially those that involved some measurement. Theorems like the one about right-angled triangles traditionally ascribed in the West to Pythagoras but independently noted in many other cultures, were essentially metrical. Classical Greek geometry started with congruent triangles that were supposed to be able to be picked up and made to coincide. The underlying approach can be said to have been tactile.
A slightly different geometric tradition was concerned with the alignments of points or the junction (appointment?) of lines, as, for instance when watchers of the night sky noticed that some of the scattered stars in the sky could be perceived as certain constellations. Greek mathematicians discovered some incidence properties of triangles such as the concurrence of the medians or altitudes. The underlying approach was here clearly also visual; and this was later developed by Renaissance artists in their exploration of the rules of perspective painting. In the hands of Desargues and later mathematicians these rules were mathematised as projective geometry
Meanwhile people continued to explore both metrical and incidence properties of triangles. For example, Euler showed that certain special points were collinear and specified their positions on the line which is now often named after him. Others were to develop this sort of work to an amazingly sophisticated extent in the nineteenth century This eventually filtered down into the sixth form syllabus of the grammar schools under the heading of "modern geometry" but along with many other traditional geometric topics it is now no longer pursued.
"Down with triangles" was the (in)famous slogan of the more formal of the post-war syllabus reformers. And nineteenth century triangle geometry did seem an old fashioned pursuit, suitable perhaps for enthusiastic amateurs but not an adequate preparation for the mathematics that was increasingly being applied in new and prestigious fields So why this present modest attempt to revive some mathematics that has been decently buried for some time? A general answer to that question might refer to a growing sense among teachers that powers of imagery need to be cultivated and that elementary geometry provides a medium in which this can be done with students of almost any age and ability.
Geometry matters - for various reasons, but also because points lines and circles are symbols of what I lie on, pass through or touch.
A more cogent particular answer would refer to the number of powerful geometry construction computer programs that are now available. Cabri-géométre for example presents the user with direct images of the basic elements - points, lines, circles - of plane geometry. You may be aware that the program will itself involve some intricate algebraic analysis, but you don't have to carry thus out yourself You can summon up the images you want merely by naming them. And you can then move them around the screen manually with the mouse. In moving an element you automatically also move the other elements of the configuration in which it occurs So in using Cabri you are always made aware of incidence properties. Moreover, because you control the transformations through the mouse you are co-ordinating hand and eye to combine the tactile and visual approaches historically associated with geometry.
The computer medium also raises searching issues about proof. Geometers have always had to distinguish two strands in their work: at first there has to be a creative exploration of a given situation to yield a conjecture about what might be the case; then there is the arduous and not always successful attempt to establish the conjecture deductively from some accepted starting point. In the sort of geometry discussed here the first investigative phase demands no particular previous knowledge, though it may call on previous experience whereas the later deductive phase invokes some established techniques and results (for example, besides a few notions about congruence some theorems about incidence such as those associated with the names of Pappus and Ceva). But now, with Cabri, an initial conjecture about a particular configuration can be tested out immediately by moving the elements to see whether the suspected invariance holds. Thus, suppose you have drawn the medians of a triangle and noticed that they seem to be concurrent. You then move one of the vertices to create lots and lots of triangles in all sorts of positions and the medians continue to seem to be concurrent. This dramatically increases your strength of conviction that they are. The proof of the pudding does after after all lie in the eating and Cabri restores some respectability to this ancient form of proof. Whether you then wish to create a deductive confirmation depends to some extent on your context.
Another powerful feature is that it becomes very easy to generalise any particular result. It is relatively easy to slightly modify a given configuration - new conjectures and theorems then come tumbling out. Consider, for example a triangle A1A2A3, with its medians AiGi meeting in the centroid G and its altitudes AiHi meeting in the orthocentre H. (The reader is recommended at first to draw this on paper.) Joining various points leads to a conjecture that GjHk and GkHj meet at a point on GH. Moving a vertex then suggests that this is an invariant property - and you might then want to try to prove it deductively in the classical manner. But you might become curious about other apparent incidences, and also whether it matters that G and H are special points. What happens if they are merely any points where lines through the vertices coincide? (The reader may now need to draw on screen.)
Triangle geometry provides an interesting area for geometrical exploration and at the same time a way of coming to grips with important issues about proof. The following pages present some of the classically known results which can be suitably explored on a computer screen. Figures accompany the text only as aids to the verbal description and readers are recommended to draw their own on paper or screen. Some brief and highly condensed hints towards deductive proofs are given in an appendix at the end, followed by some historical notes on the various mathematicians mentioned by name. A final glossary summarises a few of the most common techniques invoked in the proofs.
The sixteen numbered sections are as follows: