A Miraculous Proof (Ptolemy's Theorem) - Numberphile 

Part 2 - using the theorem to show some cool stuff: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-o3QBgkQi_HA.html And a bit extra on inversion: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-qmfFH1SyXP0.html

I love this host. She is incredibly eloquent, spends a lot of time on the proofs and introduces tricky concepts from Euclidean geometry (one of the most hated subjects) and make them surprisingly interesting and intuitive. More prof. Stankova, please!

Friendship ended with Pythagoras, Ptolemy is now my best friend

Me clicking on this video: no way I’m watching 30 minutes of this Me 30 minutes later: I have reaped the benefits of this new knowledge

This reminds me of a joke: A cattle farmer hires an engineer, a physicist, and a mathematician to build a fence around his livestock. The farmer would like to save as much money on materials as possible so they each come up with their own solution and report to the farmer. The engineer draws up a square fence seeing that its the most practical solution and the easiest to implement. The physicist decides on a circular fence, saying that the circle offers the highest area-to-perimeter ratio. The mathematician builds a fence around himself and defines the outside as the interior.

What an amazingly engaging mathematics instructor. A couple minutes in and I'm hooked, a simple mapping to prove Pythagoras's pappy theorem? Yup.

We need more of this professor. Absolutely love how passionate she is and how she takes time to explain the concepts.

Enthusiastic mathematicians are a delight to watch. Their passion and love for numbers just shines through.

As soon as she mentioned circle inversion I started getting flashbacks to a crazed Australian laughing maniacally surrounded by compasses

Fabulous video. Can’t help thinking how wonderful it would be to be a student of hers.

Every single host on this channel is my favorite.

Videos like these, going through an entire proof are the best!

I know it's a lot of work putting together these longer vids. So I just wanted to say that I enjoy them immensely, and thanks for all the effort you guys put in.

This is one of the most worthy 40 minutes + additional 7 minutes for me. I'm so so happy right now.

Absolutely fantastic Zvezdelina! I was transfixed what an amazing proof. - I'm a 66 year old ex school teacher and I'm still learning! Thank-you.

"The King that doesn't participate in the battle, but oversees everything" Wish my teacher made geometry this easy to understand.

This is one of my favorite videos (with one of my favorite hosts!). I had a significant assignment last year in which inversion was one of the keys to start cracking the problem, so it has a special place in my heart, and I love projective notions peaking at each corner. The one thing missing, which is certainly a more complicated concept, is that the "twin" of O truly is infinity, with the idea that there is only one infinity-point. In that sense, an horizontal line for example, approaches the infinity-point by going to the right and by going to the left, so we must think as if by going to infinity in one direction, you wrap around and appear in the other direction. By this logic, a line can be seen as a circle that passes through the infinity-point, cause a circle is a shape by which you can approach one point by going either way, and you wrap around. This finally leads to one of my favorite quotes by Ahlfors: "The last result makes it clear that we should not [...] distinguish between circles and straight lines [...] Accordingly we shall agree to use the word circle in this wider sense.", meaning all lines are just special circles.

"You can see technical proofs using trigonometry. *Ew, who likes those* " _Sameeee_

This is one of the most beautiful and interesting math results I've ever seen, thank you prof Stankova for your incredible explanation and diagrams!

*"I need another brown sheet"* -Stankova, 2020

I really like the format of these videos, I feel like I just attended an entire math lesson and understood a cool concept in a half hour without all the quirks of being in a classroom environment. I’ve been using a lot of transformations in art programs to edit stuff (granted, I don’t have to do the math every time and it’s with colors and selections rather than circles and lines) but these concepts are definitely floating around in the back of my mind and it’s good to see them explained.

I just love professor Zvezdelina Stankova - her voice, her eyes and of course her math. :)

Great video, Zvezdelina is one of my most favourite guests so far! _(They’re all good though)_ We even got to see the WR for the any% paper change!

This is the first numberphile in a while where I don't feel completely lost. More of this professor please.

Благодаря Ви, проф. Станкова, за невероятната страст и чудесни обяснения :)

I wish there were more videos with her! She's my favourite mathematician

Thank you, Zvezdelina! Even bigger fan of Ptolemy after your matchup. Thank you Numberphile for filming. I have been watching for years. This is now in my top 3 favs!

This is the best Numberphile video. Love the elegant geometrical proofs of classic geometry theorems.

"Something a 3rd grader can do" *Finally, my skills can be used.*

Ptolemy wins. MATHALITY!

The circle that goes through the center could be thought of as becoming a circle on the outside that just happens to be infinitely big, if there is an infinitely big circle then it’s edge would appear as a line, related to how translation could be thought of as rotation around a point infinitely far away.

"Beg your pardon" it was at this point I snorted my mouth full of tea and nearly choked laughing

I love this woman, she has so much knowledge and passion. I wish I had a teacher like her.

Please make more of these longer proof-based videos

Me: comes to Numberphile to learn interesting things about math Also me: learns that the 'P' is silent only because I'm lazy at english

Professor Stankova is tied with Cliff Stoll as my favorite Numberphile presenter, so I was extremely happy to see this video show up today! She always seems to blow my mind and didn't disappoint with her explanation of Ptolemy's Theorem. I feel like I was done a disservice by my 8th grade geometry class which made no mention of it whatsoever

C'mon Brady. After so long Professor Zvezdelina Stankova is so enthusiastic she herself wants a part 2. I like her explanations and allegories and geometry. It's a pleasure to watch Numberphile. And thanks for holding up that heavy camera for so long!~

Brady's Inversion: The frequency of a recurring guest's appearance is inversely proportional to the editing time that guest causes for Brady.

Thank you Zvezdelina Stankova for a magnificent performance as educator. Totally elegant and effective. Just WOW. and thanks to all Numberphile staff.

After the lion story, I kept hearing “lions” instead of “lines”.

Man, what a journey! It was all brilliantly explained too. Thanks a lot, Professor Stankova, I learned a lot!

This was expertly explained, every single point was clear, and I every revelation or new trick helping with the problem at hand felt natural. Amazing job.

“Trigonometry blehhh who likes those?” Gotta agree with that when it comes from a mathematician !!😂

Wow. Not an easy concept to understand, explained extremely well and obviously edited together very well. I’m sure this was a lot of work but so worth it! Huge fan of this one

This is one of the most beautiful proofs I've seen in a while, thanks!

This is the reason we all get superexcited about new Numberphile videos. Increadably well done!👏👏👏

Yes! The woman with the strange accent transforms elephants!

Today I learned inversion is not elephantness invariant.

Ptolemy: “Hello.” Pythagoras: “Father!?”

I’m not literate at math by any means, but this is the best explanation for why Pythagoras Theorum works.. it’s ultra intuitive and beautiful!!

At 4:05 I almost became a 5 year getting a chocolate I was bit extra excited than I should've been. But that was neat

this is the BEST way to learn and LOVE geometry. brady should be THE minister of education around the globe PS: i love you stankova

The thing that blows me away is that there is an equal number of points inside and outside the circle.

Thank you so much prof Stankova. That was quite an epic ride. Looking forward to parts 2 and 3.

I had known about the inversion and the Ptolemy's theorem as two separate pieces of knowledge. Very pleasant to see how they are connected 👍

This video is sooo good. I was tempted to stop it half way through and find all the proofs on my own because I've never had the chance to play around with inversion of circles but I couldn't help myself. You're real people video style is so good. You guys are doing awesome work.

I remember the time I discussed inversion mapping with my friends this excitedly after a complex analysis class😍 Thanks for reminding!

This is beautiful. Proving such a valuable and highly encompassing theorem already. Incredible

She’s an excellent storyteller and this presentation is captivating! Very nice.

I think these longer videos with interesting and useful proofs are better than a shorter video just about an interesting math fact

Wow! What a brilliant brilliant teacher!

Brilliant proof and a brilliant presenter to boot. Contagious enthusiasm and a wonderful way of putting the maths into its geometric context.

Inversion sounds like the guy in jail claiming he is free and it is everyone outside that is imprisoned lol

I appreciate Bradys work on putting out such an interesting video. You can notice how much effort was put into it, with all those animations on a nearly 40 minute video. Thanks

I bless the rays down in Africa

I love Zvezda 🙂

Her enthusiasm is honestly so catchy, she makes it sound like going on an adventure haha

videos like this and Epic Circles are endlessly rewatchable, I love these beautiful constructions so much

Love these longer videos with Professor Stankova's proofs! Would love to see even more.

Ever since this dropped, it has been my all time favorite numberphile video.

love Leibniz' cast face in the background

I love this. My professor tried to explain this idea to the class but I feel like it all flew over everyones heads. I had no idea what invcersion was and now it all makes sense. Thank you so much!

I love listening to this teacher.

I think this is the first numberphile that I'm going to have to watch a few times for me to be able to fully grok. (Thank you Dr Stankova. Lovely piece of math you demonstrated.)

Very nice bit on the theorem. There was a time where I found out about Ptolemys Theorem or should I say Ptolemys inequality which is the version that works for all quadrilaterals. I was obsessed with finding a simple geometric proof of it, but most proofs were too complicated for me. I also saw the inversion one but I deemed it too complicated too, there were too many formulas in the end. After days of not finding one, I gave up. But then one day inspiration hit me and I created what I think is the most simple and elegant proof of not only Ptolemys theorem but Ptolemys inequality which ist stronger. There will probably be a post on reddit and I will post pictures there but here is the idea: Ptolemys inequality: For any quadrilateral with side lengths a,b,c,d and diagonals e and f it holds: ac + bd >= ef with equality iff the quadrilateral is cyclic (Ptolemys Theorem). Proof idea: Take the quadrilateral and copy it 3 times. Scale one version by a factor of a, another by b and the last one by e or f (depending on which diagonal is inbetween a and b). Now the first version has a diagonal of length af, and the second has on with length bf, while the third has the adjacent sides with length af and bf. Now just sneek in the third version into the first and second the right way, and a triangle with side lengths ac, bd and ef will reveal itself. Now it‘s just triangle inequality. qed

No doubt the most intriguing, well-paced, and elegant Numberphile video I've yet come across.

Zvezda is always such a delight!

love this hosts passion, great drawings, great intro to the topic!! and the longer video format are so much better. please more Zvezdelina Stankova

When you're so early and the video's so long that it is guaranteed that anyone who has commented, liked, and disliked, has not watched the entire video yet

What a charismatic lecturer! I would certainly look forward to her classes. Thanks!

Brady seems to be able to bring out the best in these mathematicians.

We appreciate all the hard work Brady!!! This one was a real treat :)

This is the most awesome Numberphile video in a long time! Because of the elegant proof, prof. Stankova's engaging and likeable personality and the humour in the video. Prof.: Now we can prove Ptolemy's theorem. Brady: THANK DEAR LORD!

Inversion is a translation that I was never taught in school. It's fascinating! Excellent video and teacher.

Loved it, kept me glued to the screen the whole time

Stankova‘s presentation of Ptolemy‘s theorem is superb.

She explains things so well, love it

Really liked the detail on this episode. Prof Stankova is very engaging and does a great job of identifying the places that a bit of reinforcement will be useful to understanding.Thank you.

i need a 3blue1brown style animation for the inversion transformation

I know about inversion and about Ptolemy's theorem but your presentation is amassing and it was really a pleasure to listen to you and your colleague.

She's my favorite person on this channel

Really liked that video, long format and complete proof. Prof is very pleasant to listen to !

The people demands more videos of Zvezdelina

It's been a long time since I've been able to just sit, shut out the world around me, and lose myself in any video. What a wonderfully engaging guest!

What a amazing teacher!

This is genuinely so mesmerizing. I've just been sat here listening and I love it.

"Yes! Ptolemy would kick in!" darn, I wish my math teacher would have been that enthusiastic

What a fantastic host. I was wondering why you were complaining about editing it all until I finished and saw it was 38 minutes. She taught so well, so clearly, and so interestingly that I barely noticed that much time had passed

"our inversion is a completely different animal" love that

Just amazing explanation! wow so many concepts described so elegantly and how they are so intertwined to come up with something so beautiful