Conway's circle theorem

 In my previous post on Three Youtube videos I discussed the Mathologer video which states and gives an elegant proof of John Conway's circle theorem. In this post I want to give a more pedestrian proof that requires only simple angle chasing. Some of the video commenters appear to have found similar proofs so my contribution may simply be to present a proof accompanied by diagrams to make it easy to follow along.

For completeness I'll state the theorem. It concerns an arbitrary triangle where, in this diagram, I colour the triangle sides red, green, and blue. The triangle sides are extended by red, green, and blue lines as shown; all red segments are of the same length and, similarly the blue segments and the green segments.

Conway's theorem is that the 6 endpoints lie on a circle and, moreover, the centre of this circle coincides with the in-centre of the original triangle. As shown below.

Let's label the three interior angles of the triangle as p-2a, p-2b, p-2c (p being my typographical simplification for pi). Then, of course, 

(p-2a) + (p-2b) + (p-2c) = p

and so 

a+b+c = p.

We'll draw a containing hexagon:

You'll see that I've labelled some angles around the hexagon. Their values follow directly from the fact that the diagram has 6 isosceles triangles: their bases are the edges of the hexagon.

In the quadrilateral ABCD there are two opposite angles a and b+c, which sum to p (pi). Thus ABCD is cyclic: A, B, C, D lie on a circle. Thus the unique circle through A, B, C also passes through D. Similarly this is true for the quadrilaterals BCDE and CDEF. Hence all of A, B, C, D, E, F lie on a common circle.

The centre of this circle lies on the perpendicular bisectors of the chords AB, CD, EF. Obviously (isosceles triangles again), each bisector passes through one of the triangle vertices, bisecting the angle there. Therefore these bisectors meet at the centre of the in-circle of the triangle.

Three Youtube videos

 When Youtube was launched in 2005 I didn't expect that an internet service for amateurs to share home videos would be more than a niche interest. I was totally wrong. In less than a year it was attracting 100 million views per day and in December 2006 it was acquired by Google. Nowadays Youtube is one of the first ports of call when searching the internet with videos on topics ranging from "How to fit a mirror to a bicycle" to full-length courses on Moral Philosophy.

In this post I want to discuss three videos I enjoyed today (see how my conversion is complete!) which are loosely linked in that they challenge mainstream societal ideas.

The first of these is a 10 minute presentation by Zach and Kelly Weinersmith on the feasibility of creating settlements on other worlds. I came across this video by accident (a common Youtube occurrence) because I follow Zach's clever daily cartoon "Saturday Morning Breakfast Cereal". 

The Weinersmiths have just published a book "A City on Mars". Apparently they began the project with an optimism that one might summarize with "Yes, there are technical problems, but human beings are good at solving such problems. Let's look at what has to be done." However, in their video today they admitted to a complete volte-face. As a result of their research for the book they are now extremely pessimistic about whether these problems will be solved any time soon. Their conclusions are argued at greater detail in their book but this video is a punchy summary of what they found.

I'm pretty sure that scientists working on how human beings can live on Mars or the Moon are well aware of all the difficulties. However, outside that community of experts, I think people generally would presume that our aspirations to colonise other worlds are not just pie in the sky (pun intended). Who can blame them when our entertainment culture routinely serves up movies like "Don't Look Up".

So, all credit to Zach and Kelly for stepping up to fill a hole in the public understanding of science.

My second Youtube video is the latest in the excellent series of mathematics videos by the Mathologer Burkard Polster. It is very hard to make watchable mathematics videos because people's math backgrounds vary from the all too common "I've never got math. It's just not for me" through "I wish I'd paid more attention at high school and could appreciate some of the more important aspects" to "I love math but still find it hard". Polster's videos span a range within the second and third of these categories. He has an expositional style that is clear, conveys genuine enthusiasm without distracting witticisms, and uses computer animations brilliantly.

Today's video, entitled "Conway's Iris Problem", starts from an arbitrary triangle, constructs 6 additional points, and proves that these points lie on a circle. Simple hypotheses, a simple construction, and an easy to understand conclusion. What is not so simple is the reasoning towards the conclusion. We are conducted through two slick proofs supported by the usual helpful animations. 

Mathematicians often use phrases like "This is a beautiful proof" which often mystifies a non-mathematician. The great challenge for a math educator is to awaken a sense of wonder in a student when understanding a clever proof. I don't know a better metaphor than "beauty" but I know it when I see it.

Anyway this demonstration is beautiful. Polster knows it and does a great job at helping viewers experience that pleasure.

Nowhere in the video (or in any of the Mathologer videos) does Polster say that what he is describing is useful or economically worthwhile. He is concerned solely with getting across the pleasure of understanding the reason a mathematical statement might be true. When I was a young student myself this was generally the spirit in which mathematics was taught. Things are different today. It has become commonplace for teachers to motivate mathematics by appealing to its utility in the real world. As a result it is widely believed in our society that mathematics is valuable because it is such an essential tool in engineering and science. 

Of course we must not lose sight of this great utility of mathematics. But if that is the only reason we study it then something has very definitely been lost. For that reason I loved this video which celebrates the purity in Pure Mathematics.

The final video I'd like to discuss is by the physicist and philosopher Sabine Hossenfelder. I had seen some of her older videos and, while I admired her clarity of thought and precision of language, I had always wondered whether some of her negative comments about the scientific establishment might stem from career frustration. However her opening sentence in her latest video, in which she indicated sadly that her career as a physicist had transitioned into a Youtube poster, was so refreshingly candid that I was hooked immediately.

Hossenfelder gives an account of her education and career. She was a star student who thoroughly enjoyed her time as an undergraduate and it seemed to her that her intellectual life of a physicist would mirror the lives of physicists who had inspired her to begin with. However, despite having grades that a few years earlier would have earned her an academic position, she found it quite hard to make progress and embarked on a punishing life of going from one short-term research contract to another. For young scientists trained in the 1990s and later this is quite a common problem. This treadmill continued essentially till the pandemic arrived whereupon she changed her career and became a Youtuber.

During this time she came to realise that the model of academia was rather different to that of the great generation of physicists in the 1920s and 1930s and she bitterly summarizes her conclusions. A modern university department depends on research funding to survive. Permanent senior staff have to continually apply for research grants. These grants not only fund the planned research project but also come with an "overhead" component that pays for short-term researchers and contributes to the salaries of permanent university staff. Proposals to funding bodies have to be for projects that can be completed in a three to five year period and must not be too risky that they will frighten the funding body. A university department, if it is to flourish, must insist that its staff diligently seek out research funding - and this often influences how research proposals are formulated.

I was a lecturer and professor in four universities from 1970 to 2012 and watched this transition with increasing unease. I completely agree with Hossenfelder's analysis.

It seems that I am not the only one who thinks Hossenfelder's criticisms are valid. In the week since her video appeared it has attracted nearly thirty thousand (!) comments. I have been through some hundreds of these and have not come across any commenter who disagrees with her - in fact the vast majority of them echo her opinions with stories of their own.

This is a video which challenges the way universities (particularly science departments) operate. Hossenfelder has done an excellent job in bringing this to the attention of a large number of critics. It will not win her friends from the establishment but she can take comfort from the fact she has far more readers than even the most renowned senior researchers.