A Beautiful Proof of Ptolemy's Theorem.

Ptolemy's Theorem seems more esoteric than the Pythagorean Theorem, but it's just as cool. In fact, the Pythagorean Theorem follows directly from it. Ptolemy used this theorem in his astronomical work. Google for the historical details.
Thanks to this video for the idea of this visual proof: • VISUAL PROOF - PTOLEMY... . There are others that use the same three-triangle method too, but none that I've seen present a complete proof, which I believe I've done here. I have been told that this proof may have first been published by William Derrick and James Hirstein, "Proof Without Words: Ptolemy’s Theorem", in The College Mathematics Journal, Vol. 43, No. 5 (November 2012), p. 386.
Animations were created using Desmos Graphing Calculator (www.desmos.com/calculator) with the help of DesmosPlayer, a browser plug-in that I wrote to control the graph (github.com/MathyJaphy/DesmosP...) Screen capture turned that into a video; edited with iMovie.
Music: “Off Broadway”, iMovie Song ( • Off Broadway | iMovie ... ), “Acoustic Sunrise “, iMovie Song ( • Acoustic Sunrise | iMo... )

Пікірлер: 130

  • @Aiden-xn6wo
    @Aiden-xn6wo2 жыл бұрын

    This is one of the best proofs of Ptolemy's Theorem ever.

  • @codebitcookie8053

    @codebitcookie8053

    2 жыл бұрын

    Agreed

  • @Sciencedoneright

    @Sciencedoneright

    2 жыл бұрын

    I agree lol

  • @modern_genghis_khan0393

    @modern_genghis_khan0393

    Жыл бұрын

    I agreed too

  • @xvv25
    @xvv252 жыл бұрын

    A beautiful presentation of a beautiful proof. I enjoyed it immensely. My one suggestion is that you may want to mention that it was first published (as far as I know) in 2012: William Derrick, James Hirstein, "Proof Without Words: Ptolemy’s Theorem", in The College Mathematics Journal, Vol. 43, No. 5 (November 2012), p. 386.

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    Thank you! I'm glad you liked it, and I appreciate your letting me know the source of this proof. I have added a reference to the article in the video's description.

  • @affankhan7465
    @affankhan7465 Жыл бұрын

    Dazzled me for a while! This channel & this video is so under-rated. More power to you!

  • @Sciencedoneright
    @Sciencedoneright2 жыл бұрын

    The is literally the best proof I have ever seen of this theorem, like seriously this is pretty amazing

  • @theexplorer9927
    @theexplorer99272 жыл бұрын

    Your explanations are awesome!

  • @MichaelRothwell1
    @MichaelRothwell12 жыл бұрын

    Beautiful! I love the symmetry of this proof.

  • @cupatelj
    @cupatelj2 жыл бұрын

    This video is very good, thank you. You deserve more views and likes.

  • @draganminic4928

    @draganminic4928

    2 жыл бұрын

    Potpuno se slazhem, ali nema mnogo ljudi koji mogu da vide lepotu u dokazu neke teoreme.

  • @shlomi8307
    @shlomi8307 Жыл бұрын

    Your channel deserves million views my friend you are a good guy

  • @walts555
    @walts5552 жыл бұрын

    👍Good stuff! Everything in this channel is very worthwhile.

  • @mrshodz
    @mrshodz2 ай бұрын

    Love the animation and explanation.

  • @redactdead
    @redactdead2 жыл бұрын

    Fantastic as always!

  • @mike_the_tutor1166
    @mike_the_tutor11662 жыл бұрын

    Wow! Amazing work!

  • @sreekar9391
    @sreekar939110 ай бұрын

    Thanks for this beautiful, articulate and intuitive proof 🤗

  • @BetaTestingUrGf
    @BetaTestingUrGf2 жыл бұрын

    I would love more videos from you. also some longer and more in depth!

  • @mustbegross
    @mustbegross Жыл бұрын

    Finding this video when i was looking for ptolemy astronomy. Even this isnt what I looking for, I'm not disappointed and even thoroughly enjoying it 😊

  • @fengsiweifajarss2018
    @fengsiweifajarss20182 жыл бұрын

    This is the most beautiful prove for this theorem! thank you!

  • @hidgik
    @hidgik2 жыл бұрын

    One of the best background scores ever!

  • @MrEliteXXL
    @MrEliteXXL2 жыл бұрын

    I prefer it with your voice, btw nice work! Keep going :)

  • @munchwithmadhavi
    @munchwithmadhavi5 ай бұрын

    Beautiful explanation of Ptolemy's Theorem, outstanding!

  • @MathyJaphy

    @MathyJaphy

    5 ай бұрын

    Thank you!

  • @lucaokino6776
    @lucaokino67762 жыл бұрын

    can’t believe he actually used the music. so good!

  • @ankitbt2208
    @ankitbt2208 Жыл бұрын

    Loved the way of describing

  • @ankaiahgummadidala1371
    @ankaiahgummadidala13712 жыл бұрын

    Excellent video. Thanks a lot.

  • @feelingzhakkaas
    @feelingzhakkaas2 жыл бұрын

    absolutely wonderful

  • @levelupmath
    @levelupmath Жыл бұрын

    Can you please tell me what software you used for the presentation ? Thanks! Great idea btw

  • @MathyJaphy

    @MathyJaphy

    Жыл бұрын

    Thanks for watching! All my videos use Desmos Graphing Calculator plus screen capture, then iMovie to edit the screen capture video. For my first three videos, I controlled the animations by hand - starting sliders, turning expressions on and off, etc. Once the videos got too complicated for that, I wrote a javascript add-on that allows me to program these changes so they run automatically. Here's the Desmos Graph for this video: www.desmos.com/calculator/famrmmf7xs And my animation program is on GitHub: github.com/MathyJaphy/DesmosPlayer (it's got some bugs and quirks, but the instructions are all there if you want to try it).

  • @odemeprogres.2554
    @odemeprogres.25542 жыл бұрын

    No.1 proof bro , I love it so much 🤟

  • @kayakdauna2763
    @kayakdauna27632 жыл бұрын

    My god what the beautiful explanation ❤️❤️

  • @user-ke5id7ps8m

    @user-ke5id7ps8m

    2 жыл бұрын

    Good

  • @ogxj6
    @ogxj62 жыл бұрын

    I was the 667th upvote. I saved your video from darkness. You’re welcome.

  • @thiruvarasan2537
    @thiruvarasan25372 жыл бұрын

    very nice pictorial explanation

  • @LSMaths
    @LSMaths Жыл бұрын

    Really great and helpful May I know which app is used for animation?

  • @MathyJaphy

    @MathyJaphy

    Жыл бұрын

    Thank you! I get that question a lot, so I put the answer in the description. I used Desmos Graphing Calculator to create the animations, and I wrote a plug-in for Chrome that lets me program the animations, which are otherwise controlled manually by moving sliders and displaying/undisplaying the various equations. See the description for links to the code on GitHub if you want to try it. The README file has a link to the Desmos graph I used for this video.

  • @LSMaths

    @LSMaths

    Жыл бұрын

    Thanks a lot sir

  • @geoellinas
    @geoellinas2 жыл бұрын

    Beautiful theorem. The pioneering ancient Greeks!

  • @maheshbhatt8115
    @maheshbhatt81152 жыл бұрын

    You are doing amazing great

  • @monishrules6580
    @monishrules65802 ай бұрын

    Amazing,astonishing,unbelievable beautiful

  • @sokratislamprakis6431
    @sokratislamprakis64312 жыл бұрын

    I found an amazing Chanel, if only there were more videos to see..

  • @dr.rahulgupta7573
    @dr.rahulgupta75732 жыл бұрын

    Excellent proof ! Vow !!

  • @belinha915
    @belinha9153 күн бұрын

    Very Nice!

  • @mathbbn2676
    @mathbbn26762 жыл бұрын

    Look at the triangle show, it's really easy to understand if we follow his teachings really great.

  • @pool7216
    @pool72169 ай бұрын

    Hermosa demostración.

  • @Programmerdeveloper-vl1bu
    @Programmerdeveloper-vl1bu5 ай бұрын

    It was hard to believe that it could be this easy...

  • @Ni999
    @Ni9992 жыл бұрын

    Subscribed!

  • @User-54321n
    @User-54321n2 жыл бұрын

    Beautiful ❤️

  • @kenhaley4
    @kenhaley42 жыл бұрын

    Nice!!

  • @leonellopez4026
    @leonellopez40262 жыл бұрын

    Amazing video made with only four minutes!

  • @Living_for_Him_Alone
    @Living_for_Him_Alone Жыл бұрын

    Thank you ❤️

  • @warmpianist
    @warmpianist2 жыл бұрын

    This is really interesting proof. I do have concern on "scaling A" as you can't scale a segment with a factor of another length of segment this with only compass and straightedge. A*B is only interpretable as area, not length. Is there any way to circumvent this issue?

  • @warmpianist

    @warmpianist

    2 жыл бұрын

    Ok so I looked at the animation again and we can actually scale by "F/A" for second triangle and "B/A" for third triangle instead, we get C+DB/A = EF/A. And some small algebra should lead to the same Ptolemy's theorem.

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    Granted, but there's nothing that prevents scaling a triangle by a unitless quantity that happens to be equivalent to one of the side lengths.

  • @nighteyemonroe8057

    @nighteyemonroe8057

    2 жыл бұрын

    if you admit that (1,1) and (0,0) can be constructed A*B can also be constructed.

  • @QuantumHistorian

    @QuantumHistorian

    2 жыл бұрын

    No, even the Greeks knew how to draw a line segment whose length was equal to the product of the length of any other two line segments. It's the *lengths* of the lines being multiplied, so there's no dimensional problem. Coming from a physics background, I had the same confusion as you for ages.

  • @warmpianist

    @warmpianist

    2 жыл бұрын

    @@QuantumHistorian if you are given a unit length, then yes. Is it possible without using unit length?

  • @oscarsilva9644
    @oscarsilva96442 жыл бұрын

    This proof deserves handreds of thousands of likes.

  • @gandurifamily550
    @gandurifamily5502 жыл бұрын

    Thanks sir

  • @varunsohanda2601
    @varunsohanda26012 жыл бұрын

    It was awesome!

  • @saeedzargar4458
    @saeedzargar44582 жыл бұрын

    Just wow!

  • @user-goral
    @user-goralАй бұрын

    Very cool.

  • @anuamba
    @anuamba9 ай бұрын

    The animation is what helped me understand it most. How do you do it?

  • @MathyJaphy

    @MathyJaphy

    9 ай бұрын

    Hi, and thanks for the comment. I explain what I use to animate my videos in the description, along with links to the software if you want to try it yourself. Basically, it's Desmos Graphing Calculator turned into a programmable animation with a browser plug-in that I wrote which I call DesmosPlayer.

  • @anuamba

    @anuamba

    9 ай бұрын

    @@MathyJaphyBut the player plugin is deleted from github. Can you share it?

  • @anuamba

    @anuamba

    9 ай бұрын

    @@MathyJaphy Holy Fuuk you replied to 2 year old video. so can you share the link to the desmos player plugin cause it is removed from Github and thanks

  • @MathyJaphy

    @MathyJaphy

    9 ай бұрын

    I'm still making videos for the channel, so I notice comments on any of them, new or old! :-). I tested the link in the description before replying, and as far as I can tell, the repository is still there. I see my instructions and my javascript file. I'm not a GitHub expert so I don't know why you can't see it (assuming you entered the link correctly). If you have any idea, let me know and I'll investigate.

  • @MathyJaphy

    @MathyJaphy

    9 ай бұрын

    @anuamba I may have found the reason that you couldn't get to my GitHub. It looks like a KZread bug. When you click on the link in the description, it includes the close-parentheses in the URL! I fixed it by removing the period after the close-parentheses. Could you try it now and see if the link works for you?

  • @rivkahlevi6117
    @rivkahlevi61172 жыл бұрын

    Very nice!

  • @rivkahlevi6117

    @rivkahlevi6117

    2 жыл бұрын

    PS love the outro.

  • @shlomi8307

    @shlomi8307

    Жыл бұрын

    Ketiva v chatima tova sister. I also found a good channel of a friend called bethalgebra just new

  • @wesleysuen4140
    @wesleysuen41402 жыл бұрын

    v cool! (but could small letters be used instead?)

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    That was my problem when I first posted this video (LOL): kzread.info/dash/bejne/ao2jj9ird7TVoaw.html. But seriously, I assume you mean that upper case letters usually represent vertices, not side lengths. I wasn’t thinking about that when I set up the animation. I guess I preferred the aesthetics of capitals. I’ll be more careful about following convention in the future.

  • @hihello3112
    @hihello31122 жыл бұрын

    Which software you use for animation

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    Desmos graphing calculator.

  • @hanswust6972
    @hanswust697211 ай бұрын

    Indeed beautiful. 🤗

  • @NachoSchips
    @NachoSchips Жыл бұрын

    Hey I'm looking for a nice Introductory geometry book with many exercises. I'm starting my major next year and I have been focusing mainly on calc, proofs and linear algebra but I'd like to refresh my knowledge in geometry

  • @MathyJaphy

    @MathyJaphy

    Жыл бұрын

    Thanks for asking, but I don't have anything to suggest, as I'm not a math professional. I'm sure you can find recommendations on reddit, r/geometry. Good luck with your major! See you in #SoME3, perhaps?

  • @antoniopedrofalcaolopesmor6095
    @antoniopedrofalcaolopesmor60952 жыл бұрын

    I would have used small letters a, b, c, d, e, f for the measures of the line segments. Capital letters are conventionally used to denominate points.

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    You're right. It didn't occur to me until after posting the video that I was not following the usual conventions. I leaned towards capitals for legibility on small screens, but I probably could have made it work with lower case.

  • @alancruz4930
    @alancruz49302 жыл бұрын

    Wow, que satisfactorio es entender algo

  • @karelin_e
    @karelin_e2 жыл бұрын

    Hey, where are new videos? It's good content, keep work

  • @SuperYoonHo
    @SuperYoonHo Жыл бұрын

    subbed

  • @antoniopedrofalcaolopesmor6095
    @antoniopedrofalcaolopesmor60952 жыл бұрын

    Any convex quadrilateral whose opposite angles add up to 180° is a cyclic quadrilateral, right?? I mean, I have no doubt that for any cyclic quadrilateral, opposite angles add up to 180°. But is the reciprocal true?, i.e. is it true that any convex quadrilateral whose opposite angles add up to 180° is a cyclic quadrilateral?? (i.e. there is a circle passing through each of the four vertices). How can you prove it?

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    Excellent question. Yes, the reciprocal is true. By way of contradiction, assume you have a convex, non-cyclic quadrilateral ABCD where angle ABC and angle ADC are supplementary. Draw the circumcircle of triangle ABC, and note that D is not on that circle. If D is outside the circle, find the point on line AD that intersects the circle, and call it E. If D is inside the circle, extend AD until it intersects the circle, and call that point E. Since ABCE is a cyclic quadrilateral, angle AEC is supplementary to angle ABC and must therefore be equal to angle ADC. That's impossible.

  • @antoniopedrofalcaolopesmor6095

    @antoniopedrofalcaolopesmor6095

    2 жыл бұрын

    @@MathyJaphy Wow, nice! Thank you! *You meant: " If D is outside the circle, find the point on line AD that intersects the circle, and call it E."

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    Thanks for pointing out my typo. I’ve corrected it.

  • @egvijayanand
    @egvijayanand2 жыл бұрын

    Such a circle is known as Circumcircle. It's a circle that passes through all vertices (corner points) of a polygon.

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    I know. I chose to avoid using that technical term in my script. It sounds just a little bit funny. :-). Thanks anyway for the comment!

  • @user-ot3ne5zb3f
    @user-ot3ne5zb3f4 ай бұрын

    LETS SHOUT FOR PTOLEMY!

  • @benedettopagano3536
    @benedettopagano35362 жыл бұрын

    Proud to be the 69th subscriber

  • @draganminic4928

    @draganminic4928

    2 жыл бұрын

    Your joke is in a bad taste.

  • @alencherian1739

    @alencherian1739

    2 жыл бұрын

    NICE

  • @aram8832
    @aram88322 жыл бұрын

    I thought you were saying it's a sick quadrilateral, I was like maybe corona.

  • @draganminic4928

    @draganminic4928

    2 жыл бұрын

    So, did you disinfect your hands? 😆

  • @lamalamalex
    @lamalamalex2 жыл бұрын

    So…. I’m seeing a pattern of video content just leaving my sight into the edges of my screen.

  • @wafy2710
    @wafy2710 Жыл бұрын

    Insanely tought. Dictionary is licked to thank you😷

  • @MathyJaphy

    @MathyJaphy

    Жыл бұрын

    Thanks for the comment! I don't understand it, but I'll take it as a compliment. :-)

  • @ptaszor9779
    @ptaszor97792 жыл бұрын

    The video is great but would be better with voice over in the second part

  • @SabrinaHoq
    @SabrinaHoq3 ай бұрын

    Can you explain me how can you solve the factor of 'A','B' or 'F'..

  • @MathyJaphy

    @MathyJaphy

    3 ай бұрын

    I would be happy to, but I don't understand the question. What do you mean by "solve the factor of..."?

  • @SabrinaHoq

    @SabrinaHoq

    3 ай бұрын

    @@MathyJaphy it would be helpful for my project if you could provide the measurement of the sides of the triangle..

  • @MathyJaphy

    @MathyJaphy

    3 ай бұрын

    ​@@SabrinaHoq Yes, you're referring to the part where I scale each triangle by different factors. But I don't know what you mean by "solve". Some commenters have questioned whether it's okay to multiply a length by a length because the units become squared. But I'm not multiplying by a length, just a value that happens to be the same as the length of one of the other sides. You can always scale the sides of a triangle by a constant value without changing its angles. Does that answer your question?

  • @SabrinaHoq

    @SabrinaHoq

    3 ай бұрын

    @@MathyJaphy Can I use the same constant value every time,??

  • @MathyJaphy

    @MathyJaphy

    3 ай бұрын

    @@SabrinaHoq The lengths of the triangles' sides (A, B, C, D, E and F) are defined by the quadrilateral you start with. A, B, C and D are the lengths of the quadrilateral's sides. E and F are the lengths of the diagonals. For the quadrilateral that I used as a demo in the video, the lengths are approximately (A=1.49, B=1.11, C=3.34, D=1.66, E=1.98, F=1.94). Is this helpful?

  • @mhgaming7107
    @mhgaming71078 ай бұрын

    best 😉😉😉😉

  • @soshakobyan3123
    @soshakobyan3123Ай бұрын

    I cannot believe that you are not a mathematician.

  • @adb012
    @adb0122 жыл бұрын

    I know the answer to the last question (3:54): VERY!

  • @gandurifamily550
    @gandurifamily5502 жыл бұрын

    Post more

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    Soon… 😃

  • @shivansh668
    @shivansh6682 жыл бұрын

    Using technology makes math more fun!!😎

  • @SUCCESSWITHSUBODH
    @SUCCESSWITHSUBODH2 жыл бұрын

    I am from 🇮🇳India.

  • @alencherian1739

    @alencherian1739

    2 жыл бұрын

    🤨

  • @Claudius_Ptolemy
    @Claudius_Ptolemy Жыл бұрын

    Yup, your welcome :)

  • @MathyJaphy

    @MathyJaphy

    Жыл бұрын

    Thanks, I'm a big fan! Got any other theorems for me to animate? :-)

  • @ruxleec
    @ruxleec2 жыл бұрын

    Boing! ???

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    Translation…? :-)

  • @ruxleec

    @ruxleec

    2 жыл бұрын

    @@MathyJaphy The "Boing!" at the end's missing? 🙃

  • @MathyJaphy

    @MathyJaphy

    2 жыл бұрын

    Ah, got it. Yeah, that silliness was only for the joke video. I didn’t think I’d use the song again either, but I needed something for the end screen in this particular case.

  • @ernestschoenmakers8181
    @ernestschoenmakers81818 ай бұрын

    One peculiar thing is if all the opposite sides are equal then you'll get the Pythagorean Theorem.

  • @MathyJaphy

    @MathyJaphy

    8 ай бұрын

    Yes, indeed! That's what the first 30 seconds of the video is about, only in reverse. I start with Pythagoras, then generalize to the case where opposite sides are _not_ equal. :-)