A Miraculous Proof (Ptolemy's Theorem) - Numberphile

Featuring Zvezdelina Stankova... Want more?
Part 2 (bringing in Pentagons and the Golden Ratio) is at: • Pentagons and the Gold...
More links & stuff in full description below ↓↓↓
Zvezda's Numberphile playlist: bit.ly/zvezda_videos
Zvezda's webpage: math.berkeley.edu/~stankova/
Epic Circles: • Epic Circles - Numberp...
Triangle Magic Highway: • Triangles have a Magic...
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[@numberphile]

Part 2 - using the theorem to show some cool stuff: kzbin.info/www/bejne/pWS0c5qhhs6SfqM And a bit extra on inversion: kzbin.info/www/bejne/p57Jd3tniN6LhpI

[@PhilBoswell]

I like the extra length, gives us more to chew on!

[@HighMansx]

Name typo! It's Zvezdelina Stankova, missing the 'Z'.

[@kasajizo8963]

Yo Numberphile we need some more Simon Pampena

[@LindaRistevski]

You made my day, Numberphile! Quality!

[@qwertyuoip1234]

Man, long shoot!

[@sphyr4973]

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!

[@TheAradune]

The only access i have to math is geometry. If there is a visual, it is more likely that i can understand it or at least get a grasp why it could be useful.

[@mrk4022]

It's hated right, but to some it's the most beautiful thing in the world. I love it.

[@xpqr12345]

I'm not that picky: as long as I get Numberphile videos, I'm happy. So just give me more Numberphile videos....

[@marios1861]

@@TheAradune when you stop trying to USE everything a new world will open up to you. and this is rich coming from an engineer lol.

[@steamer1]

Yes. More.

[@superj1e2z6]

Friendship ended with Pythagoras, Ptolemy is now my best friend

[@Pyrostar0]

I see what you did here

[@lyrimetacurl0]

Lol

[@CalamityInAction]

Pythagoras was a joke. He had a rational number cult, maybe had a fear of beans, and he didn’t even think of the Pythagorean theorem

[@bsharpmajorscale]

@@CalamityInAction Don't forget that time they killed the dude who proved that root 2 was irrational.

[@asherrajah1832]

@@bsharpmajorscale literally came from that video

[@carsonfang5540]

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.

[@b.griffin317]

oy vey!

[@CraftQueenJr]

This is like the red rubber ball joke.

[@tlf4354]

I'm an engineer, and I only want to say that the mathematician should be careful, if he stands right in the center of the fence he would be redefined as existing at the edge of the universe....And worse of all, i mean ALL the edge of the universe....

[@icollectstories5702]

And maybe safe from lions, too!

[@victos-vertex]

@@tlf4354 I think this isn't true. Not only did the mathematician - not - use an inversion but simply a redefinition, on top of that the ground of the earth is considered the plane he's working in, not the entirety of space as none of the stock doesn't live on the Earth. However the Earth can be considered a sphere and where do the lines projected from the center intersect then? Well they do so exactly on the opposite side of the globe. Let's assume the stock lives anywhere but the north pole and you want to build a fence around them. The mathematician could just go to the north pole, build a fence there and then declare everything inside that circle as the outside. If he however used an inversion then the center of the north pole would be exactly mapped to the center of the south pole. This is easy to imagine (and show) if you just take a tennis ball and put a ring on it. It shows that in case of a sphere what's inside or outside "of the circle" is pure definition as any circle divides the sphere into two areas of circular shape. A small circle around the north pole could just be a very big circle around the south pole.

[@quartzofcourse]

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

[@lxathu]

Indeed, I also had the fear of a long story BEFORE it started.

[@collingwoodfan72]

YES!!

[@danielconnor8516]

@Mr. H yes, but it's a nice way to be introduced to inversion. Like a new tool sitting there waiting for you to use it somehow.

[@dayceem]

Nope. 30 minutes in, it's apparent that I've flunked grade 3.

[@thecallankids4718]

"I have reapdeffects" of this knowledge

[@joshhickman77]

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

[@24jh42]

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

[@danielsantrikaphundo4517]

I agree with on the first part; on the second part, remember math =/= numbers.

[@fgc_rewind]

math isnt about numbers

[@mattiasselin4955]

I love George R.R. Martin's love for letters

[@peterroemer1]

Amen!

[@PC_Simo]

@@fgc_rewind If you think Math is about numbers, you probably think that Shakespeare is all about words. You probably think that dancing is all about shoes. You probably think that music is all about notes. 😃

[@user-jc2lz6jb2e]

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

[@TamaraTkacova]

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

[@laurisgatiszarinovs2891]

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

[@mrk4022]

Trigonometry is not true geometry. It's even taught within algebra course.

[@shantanunene4389]

Synthetic is love

[@aatmansupkar8747]

This guy speaks wisdom.

[@shantanunene4389]

@@aatmansupkar8747 Hi

[@aatmansupkar8747]

@@shantanunene4389 Hii

[@tomkirbygreen]

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

[@knugie]

You are, now :-)

[@jeromesnail]

Every single host on this channel is my favorite.

[@centralprocessingunit2564]

parker square > any theorem

[@peterevans3310]

I feel the same.

[@omeraydindev]

so you don't like married hosts???

[@mdkc]

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

[@alexismandelias]

Ah that nice problem you solve by drawing 50 circles and inverting them all. Amazing

[@shrinivasabhat4628]

@@alexismandelias which problem?

[@TrackpadProductions]

@@shrinivasabhat4628 I think they're referring to the "Epic Circles" video.

[@urble]

@@TrackpadProductions man that video sure was epic

[@anteconfig5391]

@@urble ooh, that must be where I heard of this circle inversion thing.

[@steamer1]

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

[@anilsinha4968]

Yeah, some people love jargon more than the real stuff.

[@SlingerDomb]

Gotta admit that this subject is one of the hardest to teach. And it is the reason why we use algebra regularly.

[@urieldaboamorte]

@@anilsinha4968 would you like your teachers to speak in logical sentences instead of using natural language? let people learn in peace

[@sandorrclegane2307]

I really like this teacher here, but the line you chose to quote says absolutely nothing about geometry

[@quahntasy]

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

[@LindaRistevski]

Might need this as a t-shirt 😂🙌

[@kernoelnerd]

Production costs for a Numberphile video starring Prof. Stankova are about twice as high as the ones with the other Numberphile stars.

[@votepochitasupremacy9174]

Stanky leg

[@januszkubala9538]

colorized

[@kayleigh8069]

@@votepochitasupremacy9174 HAHA

[@stephenhicks826]

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.

[@johnredberg]

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

[@ipudisciple]

It therefore follows that every guest needs the same amount of editing time per year.

[@jeremymcadam7400]

@@ipudisciple we've come full circle

[@anusheelsolanki1]

For a moment I thought, thats some new mathematical theorem, I didn't know about

[@muesk3]

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

[@SlingerDomb]

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

[@samimas4343]

Yeah. It was like a thriller film where things are all around the place at first but they all come together nicely at the end.

[@erichobbs4042]

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

[@spracketskooch]

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.

[@Jako1987]

Ptolemy wins. MATHALITY!

[@mrk4022]

Easy to be right 500 years later. Pythagoras is still my favorite.

[@geez6666]

Lol

[@pcfilho425]

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

[@StefanReich]

Not a fan

[@CaslavNorisavljevic]

My favorite professor too

[@dudhauwhejrnrk]

OK creep.

[@steffen5121]

@@dudhauwhejrnrk Don't act like you have no weaknesses.

[@manuc.260]

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.

[@thinkingahead6750]

I really enjoyed this not simply for the classical geometry, a subject that is so delightful if taught well but for the sheer enjoyment conveyed by Zvezdelina Stankova. People are so lucky to be able to see this in their formative years., I wish I was seeing this 50 years ago.

[@swfreak258]

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

[@zs9301]

Please make more of these longer proof-based videos

[@johningham1880]

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

[@Subpar1224]

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

[@Sharklops]

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

[@zidanez21]

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

[@mrk4022]

Why weren't you sleeping at 4:05, Zinedine? Joke, loved that little bit too but so much more was yet to come!

[@seanp4644]

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!

[@costaran]

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

[@mr.johnson3844]

Unfortunately, if all education was implemented this way, there would be a lot of gaps. Understanding the basics sometimes requires tedium and that's unavoidable.

[@lxathu]

He kinda is already. And he has also one shared proof. He, along with a lot like him, proved that however far away from school, people are thirsty for science. Not all, but a respectable amount.

[@rulersonicboom4737]

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

[@lineikatabs]

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

[@improve_zew1er356]

ты понимешь по англиский? я не понял на англ, а на русском да)

[@banjofries]

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.

[@MrSasyB]

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

[@jonopearson83]

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

[@rv706]

Why? (I'm not a native English speaker, so I'm curious to know why; does it sound old fashioned?)

[@urieldaboamorte]

@@rv706 This expression is usually used when the other person is saying something ridiculous or outrageous. So, in this context, she was "overreacting" to someone complaining about the function.

[@ubertoaster99]

@@rv706 Yes.

[@yimoawanardo]

XDDD

[@urieldaboamorte]

@@ubertoaster99 I don't think it sounds old-fashioned, just overtly formal

[@mueezadam8438]

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!

[@hindigente]

Today I learned inversion is not elephantness invariant.

[@stephenhicks826]

Love this comment!

[@ipudisciple]

Technically, you learned that elephantness is not inversion invariant. The grammar is like "length is rotation invariant" or "angle is scale invariant". I'll show myself out.

[@hindigente]

@@ipudisciple I stand corrected.

[@GaryFerrao]

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!~

[@rstriker21]

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.

[@seijirou302]

This is how it's explained in the other inversion video linked in the description. That one is my favorite numberphile video of all time.

[@skurbanvintr0]

think you !!!

[@skurbanvintr0]

facts I was thinking the same !

[@Robisquick]

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!!

[@GodlessPhilosopher]

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

[@wash3141]

Isn't that GREAT!

[@JSole-bd1zg]

You are right (as long as we ignore the center, which maps to infinite points in the infinity! ;)

[@irrelevant_noob]

GodlessPhilosopher i feel you'd be even MORE blown away by the fact that there are an "equal" number of points in the plane as in a small segment. ;-)

[@mikewagner2299]

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

[@YourCritic]

I could listen to Professor Stankova for hours. There's something fascinating about the way she explains mathematics. Brady, you have to include her in more videos, please!

[@ckb1137]

I bless the rays down in Africa

[@gregoryfenn1462]

Roger The Shrubber I see what you did there

[@sillysausage4549]

Rains.

[@ckb1137]

@@sillysausage4549 I think you may have missed the joke

[@Grizzly01]

@@sillysausage4549 No, it is now rays. Official.

[@MartinPoulter]

An amazing journey- excellent explanation, animation and editing!

[@numberphile]

thanks

[@zincminer]

love Leibniz' cast face in the background

[@GeorgePlaten]

And one of Cliff Stoll's Klein bottles!

[@MrQbee87]

Leibniz in carbonite.

[@FranFerioli]

Looks like Leibniz is following the proof.

[@frobeniusfg]

@@FranFerioli Leibniz approofs )

[@SubtleForces]

@@MrQbee87 I had to literally laugh out loud in my Montreal apartment since I had just made another reply about Ptolemy saying "I am your father" to Pythagoras and a comment about the bust myself. I'd be stealing from another commenter by saying we have come full circle.

[@Supremebubble]

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

[@FerousFolly]

This is kinda cool actually

[@quadrannilator]

Thank you very much for sharing this particular inversion technique! It is really simple and elegant and obvious to a third grader!

[@supermarc]

Let me know when you have some pictures!

[@ipudisciple]

That's a very neat proof. Thank you.

[@LindaRistevski]

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!

[@mu-maths2778]

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

[@robertfelton8374]

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

[@basteagui]

he's got child like wonder. child like everything to be honest.

[@grmph]

She's my favorite person on this channel

[@TheJadeknight7]

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

[@chrisf84]

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

[@OlliWilkman]

The 'P' in Ptolemy and pterodactyl etc. is silent in English, but it's pronounced in the original Greek, and most other languages that borrow it. A lot of Greek has been simplified for English for some reason; Plato, Aristotle, Ptolemy… instead of Platon, Aristoteles, Ptolemaios which would be closer to the original.

[@Ezullof]

Linguistic evolution is mostly driven by lazyness and fashion anyway

[@ChristopherMeadors]

Also, spiral winged aircraft are called helico-pter.

[@GirGir183]

you mean helico-ter?

[@frechjo]

Phonotactics.

[@arwinraj861]

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

[@sanelprtenjaca9776]

Tottaly!

[@mal2ksc]

I raised my hand. I *like* trigonometry and still use it on a daily basis, although I generally do this by using only angles which have tidy sines and cosines, like 45 or 60 degrees.

[@aaabbb-lq9cp]

I love listening to this teacher.

[@Sanjay-ub7eq]

Yeah me too!

[@roger1561]

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

[@benhood7680]

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

[@SA38178]

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 👍

[@zaphodbeeblebrox5511]

Yes! The woman with the strange accent transforms elephants!

[@jeremymcadam7400]

into Aladdins shoe no less

[@dominatriex]

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.

[@lvander877]

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.

[@nvgirishs]

Wow! What a brilliant brilliant teacher!

[@slerk9]

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

[@Adamreir]

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

[@ryanbrooks2434]

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

[@drpeyam]

I love Zvezda 🙂

[@MrMomoro123]

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

[@jorn-michaelbartels9386]

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

[@isaactfa]

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

[@LaGuerre19]

36:52 "It was that easy." Lol, great video; watched the whole thing and it was worth the wait!

[@nicolaslondero9752]

The people demands more videos of Zvezdelina

[@luismaro14]

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

[@andreassumerauer5028]

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

[@user-vx9zs8zt4k]

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

[@oldcowbb]

i need a 3blue1brown style animation for the inversion transformation

[@centralprocessingunit2564]

faxx!!!!!!

[@DustinRodriguez1_0]

The ideas of similar triangles and ratios of sides are literally the only thing you need to basically derive all of trigonometry yourself. When I was in like 8th grade, I wanted to play with 3D graphics (this was in like 1991 or 92, there were no 3D GPU cards for PCs) and I ended up doing a bunch of geometry and coming up with a whole bunch of useful ratios so I could figure out where to put points on the screen based on 3D coordinates. Then the next year in school I had trig and found out those ratios were called things like sine and cosine and were even more useful than I thought!

[@worldnotworld]

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

[@sierra.tango.24]

Can we get a interview/podcast episode with professor Stankova, please?

[@AlKaBen]

I felt like a 5th grader again I love videos with her Such a great teacher!

[@chicchi1682]

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

[@route66math77]

There's a beautiful sense of wonder and discovery that highlights the joy of problem solving. I find myself coming back to this video again and again. In the realm of pedagogy, Dr. Stankova's methods are masterful.

[@lowercaserho]

Zvezda is always such a delight!

[@falpsdsqglthnsac]

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

[@SubtleForces]

In reality, it went: Ptolemy: " I am your father"

[@TimFSpears]

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.

[@sirmexicanelmo]

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!

[@gabor6259]

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!

[@danparish1344]

She explains things so well, love it

[@marcwatt]

I'm absolutely transfixed, mesmerized, and loving this presentation.... My new favorite math tutor. Thanks great presentation

[@skimmington]

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

[@MysterX79]

Went straight to my all time favorites. (after watching the whole 38 minutes!)

[@TheStillWalkin]

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

[@yaseen157]

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!

[@SpencerTwiddy]

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

[@Lord_Volkner]

AHHH! I wanted to see what would happen to a line that went through the circle, but not through the center.

[@scottriley5141]

I was wondering this as well. Maybe some kind of curve that shares the points on the circle?

[@aadfg0]

It becomes a circle through the original center and the 2 points of intersection.

[@rstriker21]

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

[@pythagorasaurusrex9853]

That's why mathematicians have no problems in jail. Definition: I am outside.

[@onemightsay248]

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

[@costicaCJ]

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.