Thales’ Theorem and Lockhart’s Lament

The Yorkshire Branch of the Mathematical Association recently hosted a talk by David Acheson entitled Proof, Pizza and Guitar. (By the way I’ll be giving a talk on card cheating for the YBMA on Wednesday 8th February at 7.30pm in School of Mathematics. All welcome but a small charge of a pound may be made.)

During David’s talk he gave a proof of Thales’ Theorem. This is a theorem that states the following. For any point in a semi-circle, the angle formed by the lines from that point to the two edge points of the base is right-angled.

Thales' Theorem

This is a good theorem in that, to me at least, it does not seem intuitively obvious (what, it’s always a right angle? Really?) and yet it is easy to convince yourself it’s true by doing some examples.

Thales (c624BC – c547BC) is often considered to be the first scientist because he was the first person (we know of) who looked for non-supernatural reasons for phenomena. Rather than believing lightning or earthquakes were caused by gods he considered more natural explanations. However, his solution to the latter involved the way that the land floated on the sea, i.e., he was totally wrong but here it is the concept of avoiding invoking the gods that counts. In the case of mathematics he is credited with a number of theorems and the main point is that, allegedly, he provided proof.

He is also credited with measuring the pyramids in Egypt. His method is interesting because it does not involve a brute force use of measuring instruments, i.e., get out measuring rods and send people up the pyramids with them. His proof is more elegant than that. He measured the height of a slave and when the sun was such that the length of the shadow of the slave was equal to his height they measured the length of the shadow of the pyramid. From this shadow the height of the pyramid could be found.

Thales' measurement of pyramids

Thales' measurement of pyramids

Returning to Thales’ Theorem, its proof is rather simple once one accepts that a triangle has angles adding up to 180 degrees and that an isosceles triangle has two equal angles. (The former statement is probably the most well known theorem in the world. I think it beats Pythagoras’ Theorem because, although most people have heard of that, they usually mis-state it.)

Let’s see what I will call the traditional proof. From the centre point draw a line to the alleged right-angle point to get two triangles. Both are isosceles as, obviously, edges from the centre have length equal to the radius of the circle.

Proof of Thales' Theorem

As in the diagram below we can label the angles with \alpha and \beta . Using the fact that angles up to 180 degrees we get that 2(\alpha + \beta ) is 180 degrees, i.e., \alpha +\beta is 90 degrees as required.

Proof of Thales' Theorem

This is the proof I use in my geometry lectures and was the one presented by David Acheson in his talk. He also said that he once overheard someone say that it’s easier than that, you just need to use a rectangle. The idea is that you rotate the triangle 180 degrees about the centre. David wondered if the person was in fact confusing this proof with a proof of the converse of Thales’ Theorem.

Anyhow, this idea led to a lot of discussion after the lecture. Some of us tried to make the rectangle argument work. Rotating the triangle produces a parallelogram (since we rotated by 180 degrees the opposite sides are parallel). The diagonals cross in the centre and have the same length. Such a parallelogram must be a rectangle, hence the angle we are interested in must be 90 degrees.

Of course, we all know and can accept that such a parallelogram must be a rectangle, but the problem is that to prove it rigorously seems to be more involved than the traditional proof above of Thales’ Theorem. In fact I’ve not been able to give a simple proof of the parallelogram result that did not involve something similar to that proof.

This rectangle argument for Thales’ Theorem was naggingly familiar to me and after I had left the talk I recalled that I had seen it in Lockhart’s Lament. This is a document circulated on the web by Keith Devlin a few years ago in which Lockhart laments the state of mathematics education in the USA. It is well worth a read if you have never seen it. However, it makes the claim that the rectangle proof is fantastic.

Unfortunately, Lockhart compares the rectangle proof not with the traditional proof above but with a two-column type geometric proof. This type of proof used to be the favoured method of teaching mathematics. Indeed it does teach rigour effectively but as a downside it squeezes all the fun out of geometry. To compare this type of proof with the rectangle proof is very unfair – from the enjoyment perspective any proof has got to be better than a two-column proof!

So, my question is, can anyone prove the “parallelogram with equal diagonals is a rectangle” result without resorting to an argument like my traditional proof of Thales’ Theorem?

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