In episode 3 I forgot to mention that in episode 2 we had made many references to ‘points on the number line’, e.g. that the point marking `\sqrt{2}`

was missing from the (rational) line. The proof of Lemma 1 (episode 3) in the proof of the centroid (as where the three medians meet) made implicit use of the number line.

## Constructions and proofs

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).

## Proofs for ‘Lines and Points’

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.

## Correction to slides (Episode 3 Lines …)

There was an error on slide 8 (‘Similar, not congruent’) of the handout some people took after episode 3 yesterday. It’s been corrected in the two versions of the slides on the webpage. Let me know by email if you have any difficulty downloading the slides, or trying the constructions by drawing or by Geogebra (which is definitely fun to learn and to play with!) – you can just google it and download it.

## Surprises all round in Episode 3

In most of our episodes I am showing some results which are ‘well known’ to mathematicians but may come as an interesting surprise to others. We shall be looking at four collinear centres of a triangle that divide their common line in a pleasing ratio. The ‘common line’ is the Euler Line marked in red in the nearby diagram. This is a classical result (18/19C). I had a surprise myself while preparing this episode to discover there had been a substantial renewal of interest in these (and very many other) ‘centres’ of a triangle in the 1980s – find out more in Tuesday evening’s episode!