Heron's Formula Proof (the area of a triangle when you know all three sides)

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  • Опубликовано: 30 окт 2024

Комментарии • 471

  • @BrainGainzOfficial
    @BrainGainzOfficial 4 года назад +390

    I've always used the law of cosines to prove it, but this is pretty slick! Thx bprp

    • @thereaction18
      @thereaction18 4 года назад +32

      Did they even have the law of cosines when Heron proved this?

    • @BrainGainzOfficial
      @BrainGainzOfficial 4 года назад +35

      The Reaction - No, but I believe his argument was purely geometric rather than algebraic.

    • @thereaction18
      @thereaction18 4 года назад +31

      @@BrainGainzOfficial I completely overlooked that they might have not even had algebra either. It would be nice to see how he did it.

    • @BrainGainzOfficial
      @BrainGainzOfficial 4 года назад +10

      The Reaction - check out chapter 5 of journey through genius by William Dunham. I think you can find it online for free. It’s a pretty interesting proof!

    • @IvanRandomDude
      @IvanRandomDude 4 года назад +3

      @@thereaction18 How do you mean they didn't have algebra? They obviously had it, at least geometric algebra

  • @carlkohweihao9584
    @carlkohweihao9584 4 года назад +99

    When you kept playing with the factorization rules at around 6:00, I already figured out how to prove Heron's formula. I tried to verify the formula years ago using the sine rule (A = ½bc sin t), but the equation got very complicated until I didn't know how to simplify it. This video shows the importance of mastering algebra, especially when it comes to solving simple problems like this.

  • @arsilvyfish11
    @arsilvyfish11 4 года назад +75

    Thats a nice proof without any trigo involved making it clean and simple😄

    • @musical_lolu4811
      @musical_lolu4811 7 месяцев назад +2

      If you look closer, you're actually indirectly proving the trig stuff (especially the cosine rule) along the way, you're just not explicitly stating the identity.

  • @baselinesweb
    @baselinesweb 11 месяцев назад +4

    Your tone is really great - that is half the battle of being a good teacher. Great video I enjoyed it.

  • @akshatjangra4167
    @akshatjangra4167 4 года назад +198

    "HE RUNS" formula
    RUclips's captions in a nutshell

    • @aashsyed1277
      @aashsyed1277 3 года назад +1

      i agreeeeeeeeeeeeeeeeeeeeeeeeeeee

  • @debblez
    @debblez 4 года назад +466

    I love how he pronounces “cancelled” as “canceldid” so much

    • @mohamedsamsudeen7694
      @mohamedsamsudeen7694 4 года назад +34

      Cancelled it😊

    • @Kitulous
      @Kitulous 4 года назад +20

      splendid canceldid!

    • @Kitulous
      @Kitulous 4 года назад +6

      @Yosif Abbas and I can't believe I actually posted that

    • @tomatrix7525
      @tomatrix7525 3 года назад +20

      As a non native english speaker, (chinese for that matter, very different) he is thinking of cancelled as the base verb, and adding ed to make it past, but he is making a past tense go into like a double past tense, so he says cancelleded

    • @jostromp7380
      @jostromp7380 3 года назад +7

      2:42

  • @diffusegd
    @diffusegd 4 года назад +51

    I got asked to do this as an interview question
    It took some, to say the least...

  • @GoingsOn
    @GoingsOn 4 года назад +8

    I’ve been wanting to see a proof of this formula for a while now. Thanks for showing this great proof!

  • @cosimobaldi03
    @cosimobaldi03 4 года назад +30

    I've come up with a formula for the area of triangles using hard algebric geometry. It takes the sides squared as inputs, so it works best on a carthesian plane.
    A,B,C are sides squared
    A=1/4 * sqrt(- A2 - B2 - C2 + 2(AB+BC+CA))
    it uses pretty big numbers so it's better to use a calculator or use it in a program... But I'm sure it can be transformed into heron's and viceversa.

  • @anisppeaks2736
    @anisppeaks2736 4 года назад +39

    I'm a backbencher sir,but your every explanation is just so easy to understand ♥️

  • @KaviAmanTenguriyaShaurya
    @KaviAmanTenguriyaShaurya 4 года назад +6

    The formula for area of quadrilateral was shocking.
    Wow! Good information.
    You are doing good.

  • @niyazikoken8836
    @niyazikoken8836 4 года назад +5

    Thats great video i always thought this theorem was long and needed so much effort so i never been curious about it and rarely used it but you changed my mind
    keep up the good work

  • @yaleng4597
    @yaleng4597 4 года назад +46

    10:38 Never heard of that, but COOL!

    • @_.Infinity._
      @_.Infinity._ 3 года назад +2

      You know the one who is credited with the invention of zero is Aryabhatta, but this dude (Brahamagupt) was the one who first gave rules to actually use zero for calculations. His formula shown here is one of the first applications of setting the other side equal to zero to solve a problem. He also has contributions in fields like linear algebra, trigonometry and astronomy.
      Here's a link to his wiki page if you're interested in knowing more: en.wikipedia.org/wiki/Brahmagupta

  • @Sci-Fi-Mike
    @Sci-Fi-Mike 2 года назад +1

    I proved Heron's formula a few years ago with SOHCAHTOA. This proof is much nicer and more concise. Great video, BlackPenRedPen!

  • @Muhammed_English314
    @Muhammed_English314 4 года назад +4

    I've squared the second quantity under the root and struggled with the algebra but finally I looked at what I had which is a fourth degree polynomial in terms of "a" and solved for "a" squared and took the square root and rearranged the solutions to get the product of the final 4 quantities Really amazing problem that I can actually solve.

  • @remopellegrino8961
    @remopellegrino8961 4 года назад +3

    Presh Talwalkar's fans will be complaining of you not using Gougu's theorem :-)
    Anyway, you are the king of RUclips math-teachers!!

  • @noahtaul
    @noahtaul 4 года назад +178

    Wow, I'd never heard of Bretschneider's formula at 10:38, that's weird! How do you prove it? It reduces to Heron when d=0.

    • @sx86
      @sx86 4 года назад +13

      bretschneider?

    • @Macieks300
      @Macieks300 4 года назад +30

      @@sx86 generalized Brahmagupta's = Bretschneider's

    • @erikkonstas
      @erikkonstas 4 года назад +14

      TBF, it doesn't exactly reduce to Heron's formula because of the way θ is defined (it would be undefined).

    • @noahtaul
      @noahtaul 4 года назад +18

      Έρικ Κωνσταντόπουλος Well it doesn’t matter what theta is because d=0 kills the cos^2(theta) part.

    • @erikkonstas
      @erikkonstas 4 года назад +15

      @@noahtaul It does, you can't cancel an undefined part in your expression just by multiplying it with zero. Instead, the whole expression becomes undefined. It's similar to e.g. 0*1/0, it doesn't equal 0 or 1, it's undefined.

  • @shawnclifford
    @shawnclifford 8 месяцев назад +1

    I was just curious as to how this was derived and this derivation is neat!

  • @dhruvvraghu6226
    @dhruvvraghu6226 4 года назад +14

    I'm so happy I found this, stay safe

  • @hipparchos
    @hipparchos 4 года назад +24

    The formula is introduced in Heron's book Περί Διόπτρας, where he proves it by using the inscribed circle, an elegant geometrical proof

    • @johnbutler4631
      @johnbutler4631 2 года назад +1

      That's the proof that I was hoping he'd do.

  • @mohammadfahrurrozy8082
    @mohammadfahrurrozy8082 4 года назад +8

    Blackcursorwhitecursor

  • @alexismisselyn3916
    @alexismisselyn3916 4 года назад +1

    I never knew about this formula, and the proof is really easy but I found this video extremely entertaniing

  • @Ramkabharosa
    @Ramkabharosa 2 года назад +2

    In any Δ ABC, the Cosine Rule gives cos(C) = (a²+b²-c²)/(2ab).
    So, sin(C)= √[-cos²(C)] =√[(2ab)²-(a²+b²-c²)²]/(2ab). ∴ area(ABC)
    =(ab/2).sin(C) =√[(2ab/4)²-{(a²+b²-c²)/4}²] which can be facto
    -rized to give Heron's formula. But who need's Heron's formula!
    For the 5,6,7 triangle; the area = √[{2(5)(6)/4}²-{(5²+6²-7²)/4}²]
    = √[(60/4)² - (12/4)²] = √[15² - 3²] = √(225 - 9) = √216 = 6√6.
    .

  • @اممدنحمظ
    @اممدنحمظ 2 года назад

    احسنتم وبارك الله فيكم وعليكم والله يحفظكم يحفظكم ويحميكم جميعا. تحياتنا لكم من غزة فلسطين .

  • @Ironmonk036
    @Ironmonk036 4 года назад +33

    Please do a video explaining the Bretschneider's formula at 10:38

    • @randomdude9135
      @randomdude9135 4 года назад +7

      Brahmagupta's***

    • @thecoolring6431
      @thecoolring6431 4 года назад +4

      @@randomdude9135 I think he was High enough..

    • @noahtaul
      @noahtaul 4 года назад +5

      Random Dude no, Brahmagupta’s formula is only for cyclic quadrilaterals, and doesn’t have the last cosine term.

    • @rasheedmohammed2227
      @rasheedmohammed2227 4 года назад +1

      No blackpenredpen spelt it wrong lol

  • @12semitones57
    @12semitones57 4 года назад +4

    Thank you! I’ve always wondered about the proof!

  • @blackscreen4033
    @blackscreen4033 4 года назад +9

    Cuuute! it's something even young students can do to really stretch their algebra skills hehe it's easy but with some algebra tricks 😊 nice

  • @sabrinashamsiddinova2639
    @sabrinashamsiddinova2639 3 года назад +1

    u r just great, thanks for making our studies easier, soon my exams, and so blessed to have found ur channel)))

  • @alterbank
    @alterbank Год назад +1

    8:26 There is a mistake here, right? The square of (a - b) cannot give you (a^2 + 2ab - b^2). It gives you (a^2 - 2ab + b^2), if you look at the identities.

    • @musical_lolu4811
      @musical_lolu4811 7 месяцев назад +1

      There's a minus right outside that distributes through.

  • @dovidglass5445
    @dovidglass5445 3 года назад +8

    Hi, on Wikipedia it says that Heron originally proved this using cyclic quadrilaterals; please could you make a video on that? Thanks so much.

    • @emmanueljosegarcia1975
      @emmanueljosegarcia1975 2 года назад +1

      It can be derived from a particular case of the generalized half angle formula. Se here: ruclips.net/video/WbkQHnNthg8/видео.html

  • @dainispolis3550
    @dainispolis3550 2 года назад +1

    By me best herons formula prof is to prof volume of equilateral triangle , and then any other treangle as resized equilateral in two directions, so this can be used as prof for n-dimensional triangles volume

  • @keertans7418
    @keertans7418 10 месяцев назад

    This video is absolutely perefect form my math project!!!!! TYSM!!

  • @stevemonkey6666
    @stevemonkey6666 4 года назад +3

    Old School Style blackpenredpen!

  • @marsbars1105
    @marsbars1105 2 года назад +1

    I actually discovered this formula in religion class by accident when I was playing around with 1/2ab * sin(C) and cosine rule (to find the angle used in the area formula and then use inverse trig identity). Thankyou for sharing this.

  • @Rkbittu
    @Rkbittu Год назад

    Very Easy to understand...Thank you

  • @manoharkanade7383
    @manoharkanade7383 2 года назад

    We can use cosine formula a2= b2+c2-2abcos(c).Work out since and use area formula A= 0.5absinc

  • @harsh.chaudhari
    @harsh.chaudhari 2 года назад +2

    This formula is actually taught in 9th grade to us, here in India, at a age when we don't know trigonometry.
    So, this is a really helpful way to understand Heron's formula

    • @0VexRoblox
      @0VexRoblox 2 года назад +3

      But iirc we didn't have the proof, just like 2 exercises from NCERT which made the formula imprint in our heads

    • @harsh.chaudhari
      @harsh.chaudhari 2 года назад +2

      @@0VexRoblox Yes exactly, no proof was given to us then

  • @alokmishra7650
    @alokmishra7650 Год назад

    I love this proof. Pls make more videos like this

  • @N1t_in
    @N1t_in 2 года назад +1

    Perfectly explained. Loved the video. Thank you so much😊

  • @uy-ge3dm
    @uy-ge3dm 4 года назад +1

    There's a simpler version. Through law of cosines, we have cos(A)=(a^2+b^2-c^2)/2ab. Then, we have sin(A)=sqrt(1-cos^2(A))=(1-cos(A))(1+cos(A)) and you can easily finish the proof using 1/2 bc*sin(A). It's the same algebra as above except you skip a lot of steps.

    • @nathanisbored
      @nathanisbored 4 года назад +1

      u8y7541 the nice thing about the method in the video is that it uses only basic algebra and no trig functions. depending on where you live, you learn this kind of algebra before you learn about trig functions (at least I did), so for that reason I would consider this method more elementary

  • @vishalmishra3046
    @vishalmishra3046 2 года назад

    Just use Sin and Cos formula and set the sum of squares equal to 1. Area = bc/2 sinA and cosA = (b^2 + c^2 - a^2) / 2bc
    1 = sin^2 A + cos^2 A = (2 Area / bc)^2 + ( (b^2 + c^2 - a^2)/2bc )^2
    (2 Area / bc )^2 = sin^2 A = 1 - cos^2 A = (1 + cosA) (1 - cosA) = (2bc + b^2 + c^2 -a^2) (2bc - b^2 -c^2 + a^2) / (2bc)^2
    So, (4 Area)^2 = [ (b+c)^2 - a^2 ] [ a^2 - (b-c)^2) ] = [ (a+b+c) (b+c-a) ] [ (a-b+c) (a+b-c) ] = [ (2s) (2s - 2a) ] [ (2s - 2b) (2s - 2c) ] since 2s = a + b + c
    Therefore, Area ^ 2 = [ s (s-a) (s-b) (s-c) ]

  • @mirkopusic1734
    @mirkopusic1734 4 года назад

    In Brahmagupta's formula I think θ is the sum of two opposite angles divided by two. I really like this video.

  • @TheMauror22
    @TheMauror22 4 года назад +1

    Great video! Please do more proofs!

  • @SeeTv.
    @SeeTv. 4 года назад +4

    1/2*a*b*sin(C) is much simpler imo. (C is the angle between the sides a and b)
    you can easily derive it geometrically if you draw h on the side a: sin(C)= opposite/hypothenuse = h/b so h=b*sin(C)

    • @erikkonstas
      @erikkonstas 4 года назад +2

      But Heron's formula doesn't need any trig at all.

  • @NoNameAtAll2
    @NoNameAtAll2 4 года назад +1

    Thank you
    I've been dreaming about learning proof of this formula some 5 years now

  • @ractan4107
    @ractan4107 4 года назад +2

    or Area=1/4 sqr((P(P-2a)(P-2b)(P-2c)) P is The perimetr of ABC

    • @jofx4051
      @jofx4051 4 года назад

      It can be used that but looks like most people uses Heron since it has simpler formula

    • @98danielray
      @98danielray 4 года назад

      okay?

    • @sergey1519
      @sergey1519 4 года назад

      My favorite ways to write it are
      (4S)^2 = (a+b+c)(a+b-c)(a+c-b)(b+c-a)
      and
      S^2 = xyz(x+y+z), where
      p = (a+b+c)/2
      x = p-a
      y = p-b
      z = p-c

  • @canman5060
    @canman5060 4 года назад

    My most favourite proof.

  • @yashwanthkumar8013
    @yashwanthkumar8013 2 года назад

    The same way i also derived this formula .....It's suprising to me that I can think like the blackpenredpen....

  • @smrtfasizmu6161
    @smrtfasizmu6161 4 года назад +2

    The first part of the proof is so simple and straightforward yet I have never been able to do it on my own (maybe I did the first part of the proof, but I know for sure that I was never able to prove this formula which bugged me since I always feel uneasy using formulas that I can neither prove rigorously or have some good intuitive understanding why they should be true without knowing the rigorous proof. Just implementing/using a formula that I have read in a textbook always felt like cheating)

    • @castilloguevaragiancarlomi6952
      @castilloguevaragiancarlomi6952 4 года назад

      You can try this formula faster knowing a little trigonometry (half angle)

    • @smrtfasizmu6161
      @smrtfasizmu6161 4 года назад

      @@castilloguevaragiancarlomi6952 I know formulas for half angles, I knew how to derive all trigonometric formulas I have been working with. But I couldn't derive Heron formula. That's what bugged me using it felt like cheating.

    • @castilloguevaragiancarlomi6952
      @castilloguevaragiancarlomi6952 4 года назад

      ​@@smrtfasizmu6161 Sorry I think I did not read your comment well my native language is Spanish

  • @anishgupta8514
    @anishgupta8514 3 года назад +2

    Using Brahamgupta formula
    i.e. sqrt(s-a)(s-b)(s-c)(s-d) = area
    Since all triangles can be circumcirled by a circle and d = 0 in a triangle
    Therefore area = sqrt(s)(s-a)(s-b)(s-)
    Hence proved
    ~anish gupta....in a hope of reply from you

  • @sabrinashamsiddinova2639
    @sabrinashamsiddinova2639 3 года назад

    I love youuu, so helpful, u just expplained it so simply and clearly

  • @Diriector_Doc
    @Diriector_Doc 4 года назад

    After doing some stuff on WolframAlpha, I got this:
    Area = (1/2) a b sqrt(1 - (a^2 + b^2 - c^2)^2/(4 a^2 b^2))
    I used a lot more trig though:
    C=cos^-1((a^2+b^2-c^2)/(2ab))
    h = a sin C
    Area = b h/2

    • @jadegrace1312
      @jadegrace1312 4 года назад

      That's the same thing

    • @Diriector_Doc
      @Diriector_Doc 4 года назад +1

      @@jadegrace1312 but a lot less simplified

  • @zoltangal3793
    @zoltangal3793 4 года назад

    Nice work. In the generalized Brahmagupta’s formula angle aplha correctly is half of sum of opposite angles.

  • @cianmoriarty7345
    @cianmoriarty7345 8 месяцев назад

    I think these might not be as clear to me as other people, because the red and black pens look the same to me because I am red green colour blind. I can't believe I have been actually watch you since you started and I only just worked out you are even using two different colour pens. I mean it's in the name! ❤️🖤💀

  • @marouaniAymen
    @marouaniAymen 4 года назад

    Great, this video proved 3 formulas at the same time, one formula attributed to a Chinese mathematician from the 13th century, then a formula found by Kahan anb finally the Heron's formula.

  • @monkklein8282
    @monkklein8282 4 года назад +2

    Why someone has to downvote educational videos like these?

  • @CDChester
    @CDChester 4 года назад +4

    What 3B1B is a patron? Damn!

  • @jayvaghela9888
    @jayvaghela9888 4 года назад

    When I was in 9th there is where I learned heron formula & as note I found brahmagupta's formula & I'm amazed that just putting d=0 you can get heron's equation.. Man Indian Mathematician were too good at that time I always love to learn more & more about them

  • @RAG981
    @RAG981 2 года назад

    Whenever I proved this formula, the class, and I, always felt that it was best to stop when we got to the stage
    A= sqrt((a+b+c)(a+b-c)(a+c-b)(b+c-a) because this is much easier to apply to any problem. Try it!

    • @RAG981
      @RAG981 2 года назад

      Sorry, I forgot to write the divide by 4!

  • @adenpower249
    @adenpower249 4 года назад +14

    My man please prove Stewart's theorem.

  • @rajabhattacharjee6584
    @rajabhattacharjee6584 4 года назад

    Sir you are very talented

  • @neutron417
    @neutron417 3 года назад +1

    Trigonometry proof is the best

  • @dainmeister
    @dainmeister 4 года назад +2

    I'm going to use this for right triangles from now on and nobody can stop me

  • @tauqeerkhan7154
    @tauqeerkhan7154 4 года назад

    Awesome explanation!

  • @alialyahyai6886
    @alialyahyai6886 4 года назад

    Nice prove

  • @Mihau_desu
    @Mihau_desu 4 года назад +1

    Hey! Great to see your proof of Heron's formula. The way I know is based on formula A = bc*sin(a) where a-(alpha) is angle between sides b and c in a given triangle. Would love to see more geometry on your channel.

  • @BozoTheBear
    @BozoTheBear 4 года назад

    Beautiful. A really excellent explanation.

  • @Anuj_906
    @Anuj_906 2 года назад +1

    ThanksMaster

  • @niom9446
    @niom9446 2 года назад

    Finally a proof I understand: :’)

  • @stefannikolov3505
    @stefannikolov3505 4 года назад

    I enjoyed very much. Thanks for making such nice videos!

  • @truthseeker3910
    @truthseeker3910 3 года назад +1

    Nice upload

  • @Rasa_b
    @Rasa_b 3 года назад

    Embezzlement can be very well demonstrated using geometry

  • @musical_lolu4811
    @musical_lolu4811 7 месяцев назад +1

    3:47 is literally the cosine rule, look closer.😂

  • @tomblackford451
    @tomblackford451 10 месяцев назад

    Your proof is much better and self-contained than Taylor Swift's!

  • @egillandersson1780
    @egillandersson1780 4 года назад +3

    I never tried to prove it, because I thought it was very more complicated.
    Maybe it is, but with you, all becomes simpler.
    Hope to see you soon at the black board back !

    • @blackpenredpen
      @blackpenredpen  4 года назад

      I have a few pre recorded videos in my usual setting. Hopefully the current situation gets better soon for everyone.

  • @yoavshati
    @yoavshati 4 года назад

    3b1b is one of your patrons? That's awesome!

  • @solcraftdev
    @solcraftdev 2 года назад

    I've proved this, its really easy, define the hypoteneuse as c=sqrt(a^2-h^2)+sqrt(b^2-h^2), A=(hsqrt(a^2-h^2)+hsqrt(b^2-h^2))/2

    • @panPetr0ff
      @panPetr0ff 2 года назад

      You dont know a value for h

  • @GlorifiedTruth
    @GlorifiedTruth 3 года назад +2

    YEEESSSS. I LOVE IT.
    Now how the hell did Heron ever figure that out?

  • @aniruddhvasishta8334
    @aniruddhvasishta8334 3 года назад +1

    The same way Heron's formula works for triangles and Brahmagupta's works for quadrilaterals, I wonder if there's a general pattern for any polygon with n sides. I assume that the proof for the quadrilateral formula comes from cutting the quadrilateral into 2 triangles and applying Heron's twice, so theoretically it's possible to derive a formula for a pentagon and so on.

  • @RiteshNEVERUNIFORM
    @RiteshNEVERUNIFORM 4 года назад

    Best maths teacher

  • @141Zero
    @141Zero 4 года назад +17

    Now do the formula for the area of a pentagon.

    • @erikkonstas
      @erikkonstas 4 года назад +1

      Dr. πm has made a video where he proves the formula for the area of a general regular polygon. ruclips.net/video/B07EgGGL6q0/видео.html

    • @angelmendez-rivera351
      @angelmendez-rivera351 4 года назад

      Έρικ Κωνσταντόπουλος It works, but it does not generalize.

    • @erikkonstas
      @erikkonstas 4 года назад

      @@angelmendez-rivera351 How does it not generalize?

    • @SB-wy2wx
      @SB-wy2wx 4 года назад +6

      Έρικ Κωνσταντόπουλος because a regular polygon isn’t a general polygon, it’s only one special case. Just like this video doesn’t prove the area for an equilateral, but some random triangle.

    • @erikkonstas
      @erikkonstas 4 года назад

      @@SB-wy2wx But a pentagon is a regular 5-gon.

  • @ethanwinters1519
    @ethanwinters1519 4 года назад +4

    At 10:38 shouldn't theta be the average of the two opposite angles, rather than the sum? That's what wolfram says, anyway. It's also the only way to get the right area for a square

    • @blackpenredpen
      @blackpenredpen  4 года назад +2

      Ah! Yes, you are correct!

    • @blackpenredpen
      @blackpenredpen  4 года назад +1

      Thanks for pointing this out. I just pinned your comment so others can see it. Thank you.

    • @mather468
      @mather468 4 года назад

      @@blackpenredpen Apparently it got unpinned

    • @merug4349
      @merug4349 4 года назад

      Bruno Moreira because op edited the comment

    • @assassin01620
      @assassin01620 4 года назад

      @@blackpenredpen Needs to be repinned lol

  • @dushyanthabandarapalipana5492
    @dushyanthabandarapalipana5492 2 года назад

    Thanks!

  • @AZ-tx5yd
    @AZ-tx5yd 3 года назад

    thank you so much!

  • @hocinetctmt
    @hocinetctmt 4 года назад

    Thank you for this great ful video

  • @UnKnown-lf7bl
    @UnKnown-lf7bl 4 года назад +1

    I am in 10th grade and this is the first video of BPRP that I understood well

  • @shanmugasundaram9688
    @shanmugasundaram9688 2 года назад

    The proof is simple and easy.A lengthy and hard proof using geometrical construction is given in the book "Journey through Genius" written by William Dunham.

  • @cftuba5313
    @cftuba5313 2 года назад

    Thank you

  • @stapler942
    @stapler942 3 года назад

    I love the phrase "invite into the square root house", I never thought of thinking of it that way.

  • @armanrasouli2779
    @armanrasouli2779 4 года назад +1

    thanks for the amazing content

  • @yassinezaoui4555
    @yassinezaoui4555 4 года назад

    Such a gorgeous proof ✔

  • @uttamdas-zf6pi
    @uttamdas-zf6pi 4 года назад

    It's really good sir.i want more mathematical proof sir

  • @the.rajkumar.sawant
    @the.rajkumar.sawant 4 года назад

    Good bro, your videos are amazing. Please try in upcoming videos to solve
    Derivative of x!

  • @tutorialstube2264
    @tutorialstube2264 4 года назад

    splendor, bro carry on

  • @gregoriousmaths266
    @gregoriousmaths266 4 года назад +16

    rip the small whiteboard :(
    ns video btw

    • @jofx4051
      @jofx4051 4 года назад +1

      It is digital board anyway (you write it on a computer)

  • @JungKookOfficiaI
    @JungKookOfficiaI 2 года назад +1

    this is insane

  • @smrtfasizmu6161
    @smrtfasizmu6161 4 года назад +1

    I always wanted to know this.

  • @solveit5153
    @solveit5153 4 года назад

    Wow amazing formula 😍

  • @TechnoSan09
    @TechnoSan09 3 года назад +2

    Whats brahmagupta's formula
    Yeah ik it gives the quad area but pls elaborate it

  • @GURUDEVCLASSES
    @GURUDEVCLASSES 4 года назад

    NICE EXPLANATION SIR. YOU ARE GREAT. #themathsgurudev