Ptolemy’s Theorem and the Almagest: we just found the best visual proof in 2000 years

2024-09-07

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We are making history again by presenting a new visual proof of the 2000+ years old Ptolemy's theorem and Ptolemy's inequality.

00:00 Introduction

04:27 Geometry 101

08:19 Applications

14:46 Ptolemy's inequality

18:34 LIES

25:35 Animated proofs

28:57 Thank you!

30:53 Degenerate Easter Egg

There are some other proofs of Ptolemy's theorem/inequality based on scaling and aligning suitable triangles. However, none of them is as slick, beautiful and powerful as Rainer's new proof. In particular, check out the animated scaling proof on the wiki page for Ptolemy's theorem (and this https://youtu.be/ZK08Z5A9xH4) and check out the scaling proof by Claudi Asina and Roger Nelson: Proof Without Words: Ptolemy’s Inequality in Mathematics Magazine 87, (2014), p. 291. https://www.jstor.org/stable/10.4169/math.mag.87.4.291

Rainer was inspired by a classic scaling based proof of Pythagoras theorem that I presented here /youtube/video/p-0SOWbzUYI

You can find a couple of full text versions of the Almagest here

https://www.wilbourhall.org/index.html#ptolemy

https://classicalliberalarts.com/resources/PTOLEMY_ALMAGEST_ENGLISH.pdf

For more background info check out the very comprehensive wiki pages on:

Ptolemy’s theorem

https://en.wikipedia.org/wiki/Ptolemy%27s_theorem

Ptolemy’s inequality

https://en.wikipedia.org/wiki/Ptolemy%27s_inequality

Claudius Ptolemy

https://en.wikipedia.org/wiki/Ptolemy

The Almagest

https://sco.wikipedia.org/wiki/Almagest

Trigonometric identities

https://en.wikipedia.org/wiki/List_of_trigonometric_identities

Cyclic quadrilateral

https://en.wikipedia.org/wiki/Cyclic_quadrilateral

The optic equation

https://en.wikipedia.org/wiki/Optic_equation

There are very interesting higher-dimensional versions of Ptolemy's theorem just like there are higher-dimensional versions of Pythagoras theorem. I did not get around to talking them today. Google ...

Highly recommended:

T. Brendan, How Ptolemy constructed trigonometry tables, The Mathematics Teacher 58 (1965), pp. 141-149 https://www.jstor.org/stable/27967990

Tom M. Apostol, Ptolemy's Inequality and the Chordal Metric, Mathematics Magazine 40 (1967), pp. 233-235 https://www.jstor.org/stable/2688275

https://demonstrations.wolfram.com/PtolemysTableOfChords/ an interactive exploration of Ptolemy's table of chords

Ptolemy's theorem made a guest appearance in the the previous Mathologer video on the golden ratio: /youtube/video/cCXRUHUgvLI

Here is a nice trick to make Ptolemy counterparts of Pythagorean triples. Take any two sets of Pythagorean triples:

5² = 3² + 4², 13² = 12² + 5², and combine them like this:

65² = 13² × 5²= 13²(4² + 3²) = 52² + 39²= 5²(12² + 5²) = 60² + 25².

Now combining the two right angled triangles 52-39-65 and 25-60-65 along the common diagonal in any of four different ways gives a convex quadrilateral with all sides integers. Note that you automatically get 5 integer lengths and then Ptolemy's theorem guarantees that the remaining side is a fraction. Scaling up everything by the denominator of that fraction gives one of the special integer-everywhere quadrilaterals. See also Brahmagupta quadrilaterals.

Here is a nice application of Ptolemy's theorem to a International Maths Olympiad problem https://www.youtube.com/watch?v=NHjtHOE1lks

In a cyclic quadrilateral the ratio of the diagonals equals the ratio of the sums of products of the sides that share the diagonals' end points: https://www.geogebra.org/m/XQr5jJQg This extension of Ptolemy's theorem is part of the thumbnail for this video.

T-shirt: cowsine :)

Music: Floating branch by Muted and I promise by Ian Post.

Enjoy,

burkard