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

I don't know why Ptolemy's theorem (and other theorems in geometry) isn't widely taught in schools, but it's definitely taught at university and Pädagogische Hochschule to teachers in training. So, who knows, maybe some younger teacher might feel like teaching Ptolemy's theorem in middle school or high school.
The issue is that geometry has fallen out of fashion. These days there is a lot more of statistic and probability theory in schools, but less geometry and analytical geometry. On the other hand, The concepts of Calculus are taught more thoroughly, while back in my days we only learned to calculate derivatives and integrals of real functions.

French swiss guy here, we had classes of algebra and geometry at school, and had to do a lot of demonstrations, many of which involved the angle at the center, Thales,... I loved that stuff, and it really framed my mind. I enjoyed that video the same as I did then. Nowadays my children do no more see any proof, not even Pythagore... it's a pity! Btw I still have the book, Géométrie plane, by Delessert

I feel so dumb.

So a new type of relativity. Unifying doesn't really mean correct. Things have connections that point to things that really are just effects, not causes. Is the point the effect or the line. That is the next step.. can we take the points and make them fluid yet keep the same area, or can we take lines and move them around. It seems it's the points.

My mother done Ptolemy.🤪

not really connectied, but I was reminded of that neat theorem that any quadrilateral can have a square shadow.

Isn't a "3D polygon" generally called a "tetrahedron"?

What about 4points on a conic section , do these relationships still hold?

A tiny bit less elementary, but to me much more conceptually transparent, proof can be obtained by applying inversion centered at one of the vertices, and expressing distances between the images of three other vertices as A/bc, B/ac, C/ab (for the unit radius of inversion). (Works in any dimension.)

Thanks for this great video. I have taught Ptolemy’s Theorem but was unaware of his inequality - nice! Also (to my astonishment) I was introduced to kinds of quadrilateral that I didn't even know existed, particularly the truly 3D kind. Nice work!

A couple of notes:
I wonder how many generalisations of Pythagoras' theorem there are? For now, I can see two: the cosine rule and Ptolemy’s Theorem.

I was slightly disappointed you didn't use the word "continuity" in your explanations of how 2D quadrilateral cases can be thought of as the limit of 3D quadrilateral cases.

Absolutely Brilliant. Thanks for wising all of us up!

When I learned about imaginary numbers and their exponentials, it became much easier to remember and derive the angle sum formulae.

Ptolemy was geocentrist, that's why he fell out of favor. Also, he was favored by the papal church, and that was a red flag for the thinkers in Renascence.

Could you do a lecture on Riemann hypothesis, Landau-Siegel zeroes conjecture and also Yitang Zhang's approach. Thanks.

Absolutely adorable and charming video about Ptolemy's theorem by Zvezdelina Stankova on Numberphile. I especially watched this video because that one taught me about Ptolemy's theorem. So the teaching is happening :)

I think a little lie was that the Almagest was in print for over a thousand years. I don't think it was in print until the fifteenth century, after the invention of the printing press, and it hasn't been a thousand years since that!

Oh, how I miss the days when such elegant demonstrations would come sooth my brain like butter on hot toast...

Student who remembers the sum rules for sine and cosine: 🤓
Student who derives them from the Ptolemy's theorem: 🗿

After preparing for competitive math Olympiads i hated geometry, this video has shown me the true light of how beautiful it is.

I'm wondering if one can deal with all these various cases by using "directed angles" instead.

I'd never seen the visual proof from this video. It's great! I also noticed that when you pronounce Ptolemy, "p" isn't completely silent as in normal English pronunciation, but a "p" without release. I wonder if this way of pronouncing Ptolemy is from German.

Which components can be extended to ellipses?

Ah, the Ptolomey I.

I don't understand why Ptolemy theorem isn't taught more widely not only in western countries but in Asia countries as well. It's a much deeper theorem than the Pythoragas theorem.

Mr. Burkard, first of all thank you so much for your Mathologer channel. It is wonderful. I enjoy each and every video and keep waiting for the next one :) I would like to share one thought I've got after watching the latest episode about the Ptolemy's theorem - suspecting it is quite trivial and seems different to me only - for which I apologize in advance. Here is what I thought: the Ptolemy's imparity can be generalized a little bit. Let say, we have a quadrilateral - any in fact. Add the diagonals, i.e. get all vertices mutually connected. We will have a number of line segments (original shape's sides and diagonals). Now split those in pairs such that the segments in each pair would not have common vertices. Then apply that naming convention you've been using in your video, where one pair would have A and a segment, the next - B and b, and C and c. Then it seems that the rule Aa+Bb>=Cc will hold despite which pair is a''s, b's or c's, that is c-pair should not necessarily be of the diagonals. And it is pretty obvious based on the proof you have presented. Then, the case of equality follows with all vertices located on a circle, and c-pair being the diagonals. Thank you so much for your time - and again sorry for being not enough educated to recognize a trivial result. Am already waiting for your next video! Best regards, Mike Faynberg

Absolutely stunning! It is a big joy for me to see such a simple proof. Going to 3D is perhaps not too unnatural if you know Desargue. But still ...

I think, the more general version of Pythagoras is the law of cosines (not cowsines as shown on the T-shirt).

Thanks again. I love to watch your videos on a Sunday afternoon with my coffee. Oh and cool T shirt

Great video! Get him to 1 million already! :)

Claudius Ptolemaeus (Ptolemy) was definetly underated ...
its actually strege this guy isent more known by ppl

Hey Mathologer, I just read a book that had a geometric proof (maybe?) of Fermat's last theorem. It seems good to me but I'm not really qualified to judge it's merits. I am curious though, maybe you'd be interested in taking a look? It's called "Geometric Approach to Fermat’s Enigma" by Leonello Tarabella

Wiki taught a neat equality (A-B)(C-D)+(A-D)(B-C)=(A-C)(B-D) as we treat a cyclic quadrilateral ABCD as 4 complex numbers.
Then here's how it relates to the cross-ratios. Let z1=AB*CD, z2=AD*BC, and z3=AC*BD. Through some experiment I find somehow the ratio (z1-z3)/(z2-z3)=λ is always a real number, i.e. they're colinear. I was so perplexed until I realized there exist a Moebius transformation to map the circle into a line. And λ as one of the cross-ratios is preserved and obviously real. It's just like the last animation that turn it into the degenerate case with 4 colinear points.
Now, of what use is this random fact? It's quite useful. Say we're given the points A, B, C, then their circumcircle (being the loci of D) can be given by a parametric equation of λ, a real number. According to said relation of the z's.

Wat?! Cowsine??? Hahahahaha.

Thank you for this great video! please check a video on my channel about magic square proof also confirm the same A*a + B*b > C*c

Wow - amazing proof. Fortunately I did learn the Ptolomey's Theorem when I was in elementary school. Unfortunately this implies that I was in elementary school a long long time ago.

Vielen Dank, auch an Reiner!

Best mathematics channel so far I came across

Sensational presentation. Just 3 points.
1) The sine notation, which makes such neat formulas, was invented by Indian mathematicians. Prolemey's trigonometry was much more cumbersome as he used the chord instead of the sinus. I am sure you know this, but maybe it is important to emphasize this to clarify what a great mathematician Ptolemy was.
2) I would recommend also the 2 books by Aaboe that were my introduction to the subject:
Aaboe - Early episodes in the history of mathematics
Aaboe - Early episodes in the history of astronomy
3) What tool do you use for drawing production?

Nice t-shirt