# Reasons to teach Thales' theorem

In a classical course on Euclidean, compass-and-ruler geometry, Thales' theorem has always had a prominent place. However, as the Wikipedia article says,

It is equivalent to the theorem about ratios in similar triangles,

which seems to be conceptually easier for most students (I don't have any hard data for that, but this is both my and my colleagues' experience). So my question is: should we teach Thales' theorem? If yes, should we expect the students to actually use this particular result (as opposed to ratios in similar triangles)?

Note that I am strongly opposed against purely pragmatic approach to teaching – I think that we should teach people many things that are not directly useful, but are an important part of our culture (Latin language being a classical (pun intended;-)) example) – so an argument like "it is not needed to solve problems, so it should not be taught" is not at all convincing.

• What "classical course on Euclidean, compass-and-ruler geometry" are you referring to? I've looked through several high-school geometry textbooks, and none of them seem to have this theorem. Apr 24 '14 at 1:17
• Well, every Polish high-school textbook. Thales' Theorem is a required part of a state-enforced curriculum. Apr 24 '14 at 5:33
• Since there are two theorems commonly called "Thales' Theorem," it would be best to make clear which one you mean in the text of your question. Your link makes it clear, but I thought you meant the other theorem until I checked the link. Not everyone will be so diligent. Sep 2 '15 at 21:59

Sadly, the time you (and your students) have available for this is finite. So one has to weigh usefulness carefully. Also, easier to understand is often paramount in teaching. So I'd go for similar triangles (wider usefulness, easier to understand).

• I'm also leaning towards exactly this approach – but I want to hear from teachers more experienced than me. Apr 23 '14 at 19:59

Everything depends on which axiomatic system you build everything.

Say,

• Birkhoff gives SAS-similarity condition as an axiom --- no need for Thales.
• In the classical axiomatic approach, Thales' theorem is proved first and the theorem about ratios in similar triangles is its easy corollary. (One can prove SAS directly, but likely it is better to do with Thales; at least the experience of few generations says so.)

NB. This reply mistakenly focused on a different Thales' Theorem. So it is not relevant to the posed question.

I would just like to point out that Thales' Theorem plays a key role in Delaunay triangulations, which are a central topic in computational geometry and meshing algorithms. Were Thales' Theorem connected to these cutting-edge topics, it might be seen as more relevant and important. How to reach high-school teachers with this connection is a significant challenge...

(Red dual graph is the Delaunay triangulation.
Image from Discrete and Computational Geometry, which invokes Thales' Theorem four times.)
Let me try to address Jim Belk's (excellent) question. What is important is the relationship depicted below: $A > B > C$. That it can be proved by similar triangles does not diminish the importance of the nonobvious conclusion.

To know that Thales' Theorem plays a role in what students see in The Lord of the Rings or The Game of Thrones could make it more relevant.

• But why use Thales' Theorem for these proofs instead of similar triangles? It seems to me that Thales' theorem is just a more complicated version of the same idea. Apr 24 '14 at 1:00
• I am addressing Thales' Theorem as described here. I now see there are several "Thales' Theorems" ... Apr 24 '14 at 1:13
• @JimBelk: Apologies, Jim & everyone, I was unaware there are two such theorems! Consider my answer a reply to an as-yet unasked question. :-) Apr 24 '14 at 1:15
• By the way, since it came up, I'd like to mention that your geometry book is absolutely fantastic. Apr 24 '14 at 1:20

If I could I would post as a comment.

Just to be the exception to your experience, I didn't even know (or absolutely don't remember) about similar triangles and I find the relation to Thales' theorem (that I do remember well) not that much obvious. For details, I'm 20 and probably been taught Thales's theorem around 11 so I may have simply forgot.