Did you have a go at drawing any of those concurrent lines or collinear points? Either with instruments, or with software (e.g. Geogebra). If the constructions always ‘work’ are the proofs necessary? What do they do for us? One difference with the software is being able to move the vertices of the triangle and ‘check’ the concurrency – like doing many experiments of drawing, or is it?
Could there be mistakes in the proofs? (I mean the intended proofs – not just any of ‘my’ mistakes!) Related to the question about ‘accuracy’ of Geogebra, it seems to me the software ‘knows’ more than expected. I could not fool it by drawing a third perpendicular bisector into making them truly ‘concurrent’ – only when it was using its own version of perpendicular bisector would it make the intersection one point (rather than three).
The last session (Episode 3) was all about triangles. Detailed proofs of most of the results were omitted in the talk because there was not time. I have, at last, posted all the proofs on the webpage for Episode 3 – you will see why we omitted them – they are a bit long! I’ll be more than happy to explain further if you have any (even the simplest!) questions (or corrections/omissions). But they are only 2 pages long (each one).
It’s hard to know quite how surprised we should be about lines being concurrent, and points being collinear, in the study of triangles (and in many other contexts). But when we do get a surprise it’s natural to ask ‘Can we prove it?’, meaning, ‘Can we prove this will always happen in any triangle?’. If it’s a while since you’ve looked at any proofs, I recommend the ‘Centroid’ one – go straight to the Theorem (page 2) and look back at the Lemmas if you need or wish. It involves two examples of reasoning about a pair of similar triangles, so it is a mixture of concepts and logic – good preparation for thinking about proofs in the next episode.
A central feature of mathematics is proof. This is not so much apparent at school – where learning a variety of concepts and techniques, problem solving, and getting answers (right or wrong!) and qualifications are rightly the main emphasis. But in study beyond school, proofs, and methods for construction and evaluation of proofs, are as characteristic of mathematics as experiments are in the natural sciences. We’ll survey a range of examples of proofs (good and bad) – both historical and modern, including proofs generated (or checked) by computer.
This next episode is on Tuesday 25th February.