A Fun Proof of Van Aubel's Theorem.

Van Aubel's theorem isn't much more than a curious geometrical construction, but the more you think about it, the more interesting it seems. There are a few published proofs out there on the internet. Most involve constructing similar triangles, but the one that fascinated me involved placing the quadrilateral in the complex plane and then doing complex algebra to show that multiplying one of the lines by i (i.e. rotating it by 90 degrees) resulted in an equation that equals zero, proving equivalence of the two lines. I have essentially used the same method, substituting abstract vectors for complex numbers, which makes it more conducive to animation.
Apologies for my terrible guess at how to pronounce of "Henri Van Aubel". I'm not French, and neither was he!
Source of the complex number proof: www.i-reposito...
Animation was done with Desmos Graphing Calculator. You can interact with it here: www.desmos.com...
Music: “Fifth Avenue Stroll”, iMovie Song ( • Fifth Avenue Stroll | ... )
Corrections:
0:59 It is more accurate to say that Van Aubel was Belgian, not Dutch. He lived and taught in Belgium, and wrote his theorems in French. I'm not sure what my source was for his being Dutch; I cannot find it now. It seems likely that he was of Dutch ancestry at least.

[@ivarangquist9184]

Very cool indeed. It's unique to see algebra presented without a single equal sign.

[@officiallyaninja]

and imo it shows why we use algebra. this video is super cool don't get me wrong, but it'd definitely easier to do this in algebra than through pure geometry.

[@henryhowe769]

@@officiallyaninja I feel like a lot of the use of this kind of stuff, is that while the algebra is very very useful for understanding that its true. The geometry gives you a very good look at why its true. And getting that kind of experience is really useful for actually solving problems. Being able to look at problems visually and see where the algebra needs to go.

[@spiralhalo]

@@henryhowe769 Both? Both. Both is good.

[@mloxard]

There is a equal sign but it is not written, he just said it

[@okboing]

As a person with super good visual understanding, this is just the way I think. I can prove some things to myself, without writing anything down. This one, however, was too far fetched for me to prove independently.

[@lonrabbit7776]

These are the types of things that made me realise how fascinating math is, but I was never intuitive enough to really understand the application.

[@dammit3048]

These are the types of things that made me realize how bad I am at math and how I still have no clue what is happening in the video no matter how many times he explains it

[@thisislukas8951]

@@dammit3048 SAAAMEEEE UGAA BUGGA UGH UGH

[@SuperMaDBrothers]

Pretty cool that you don’t need to do any math with coordinates, that’s all just encapsulated in the vector representation

[@Overhemd]

It's a lot easier though usually with coordinates and equations. For this example it could be done graphically more easily, but for more difficult examples, one would need to write down some variables

[@gabonafold9446]

VERY COOL. Everything was so good about this. The theorem itself, the method, the explanation, the animations, and the enthusiasm in your voice. I loved it

[@Lonech]

Very cool, very cool

[@ruferd]

I don't know how I just found your channel, but I absolutely love stuff like this. Great explanations, great visuals, concise. Simply beautiful.

[@jemakrol]

Math can be both cool and beautiful. My issues with it comes when I don't understand things. A pedagocal and well made video is certainly a good way of making things more understandable. Good job!

[@awoat]

Seems this is being recommended more, and I wanted to say this is very nice and intuitive. Haven’t seen this method used for many geometric proofs, but I hope to going forward.

[@zachb1706]

Vector proofs are a huge portion of my current curriculum

[@masonprice897]

This was so well done! Your voice makes it feel like a comforting cartoon or something

[@spiderjerusalem4009]

i read about this in needham's complex analysis 3 days ago. Algorithm really did read me

[@VeeTheGator]

This is incredible! A theory I didn't know existed shown and explained without any complicated lingo or prior knowledge, with graphics to make it easier to follow. Thank you!!

[@bitterlemonboy]

This is the first video I watch on my new smartphone...

[@hoangntran1]

I dont know why but i started to watch math problem solving as a way to entertain myself recently,keep it up the good work

[@ArloLipof]

Very cool theorem! Thanks for your amazing animations!

[@Theooolone]

I can’t believe this whole video is animated in Desmos, some amazing dedication!

[@scottekim]

Love it. A light yet surprising theorem with a satisfying visual proof. Now I gotta go try out Desmos.

[@noliveira1502]

Your presentation is gold!!! You make it easy to see the problem! Thank you, man!

[@checkthisoutmathematics4914]

just saw post of van aubel's theorem on my facebook feed. as a mathematician, had to see a proof. excellent presentation! thanks!

[@redactdead]

Beautiful animations and explanations.

[@pierrekilgoretrout3143]

what about generalizations, if we draw equilateral triangles on each sides or other shapes?

[@wiredlifter]

Not a generalisation but erecting equilateral triangles on the sides of a triangle and joining their centres yield another equilateral triangle whose centre coincides with the centroid of the original triangle. There are several more results, read about the Fermat point

[@baksu7897]

Umm... yes. Math gud

[@adb012]

Hi. Hair-splitting Devil's advocate nerd here: This doesn't work when the squares result in 2 pair of opposite-side squares with coinciding centers because then you don't have lines connecting the centers. In theory, it would not work with just one pair of opposite-side squares with coinciding centers, except that it is impossible. It is either both pairs or none. Prove it!

[@MathyJaphy]

Ah, a hair-splitting nerd after my own heart! Well, if one pair shares a center, then both would, else the lines connecting them would not have equal lengths. But who's to say it "doesn't work" in that case? The lines exist if you accept the possibility of zero-length lines, and who's to say that two zero-length lines are not perpendicular? :-)

[@adb012]

@@MathyJaphy Damn, beaten by another nerd using that ancient concept that 2 zero-length lines, which have no direction (or every), are always perpendicular to each other.

[@MansMan42069]

@@adb012 are zero length lines also parallel? And every intersect angle in between?

[@EllipticGeometry]

@@MansMan42069 I would argue so, yes. You can define such notions without decomposing a vector into a length and direction first. Two vectors are perpendicular if their inner product is zero, for instance. You can additionally require a nonzero exterior product, but that is only that: an additional requirement. I think it’s better to keep those separate by default, and maybe define _strictly perpendicular_ to incorporate both only when necessary.

[@wesleyc.4937]

@@EllipticGeometry Are you sure? Wouldn't a "zero length line" really be the same as a single point? For example, if LINE AB is defined with two POINTS like so: A(x, y) and B(x, y). Then, for line AB to be of "zero length", the (x, y) coordinate values of both points would have to the same ---- where, A = B, and (B - A) = 0. Because of this, any LINE of "zero length" would have to be "treated" like a 0-dimesnional (point-like) object... which makes it impossible for that LINE (or GLORIFIED POINT) to yield any 2-dimensional (or 3-dimensional) "vector-like qualities"... I would think.

[@XWA616G]

Totally cool! Who'd have thought of reaching for vectors for such a geometrical nicety? Wonder if it works for Ptolemy's Theorem (the intersecting chords with common ratio)

[@stoufa]

The coolest thing I saw today, and probably will be the coolest video I see this week, or month, or even year 😁 Math isn't always taught as such in schools, unfortunately!

[@jaelee5689]

The only sad part here is reading Henri as french would, when he is dutch😂

[@terrible1237]

It pains me

[@lysander3262]

This fills Henri with ennui

[@TTrisTV]

It kinda hurts xd

[@MathyJaphy]

I know, it pains me too. A disclaimer in the description is all I can do for now.

[@RexxSchneider]

It appears he was baptised Henricus Hubertus van Aubel. His theorem was published in _Nouvelles Corresp. Mathematique 4 (1878), pp 40-44_ in French, and titled "Note concernant les centres des carrés construits sur les côtés d’un polygon quelconque". It probably makes the pronunciation of his first name somewhat moot.

[@thwartificer]

How cool was that? VERY cool. Your voice sounds enthusiastic and the video itself is edited greatly

[@MrRyanroberson1]

It's incredible that this didn't even involve multiplication (of vectors), nice

[@SWASTIKB306]

If maths was taught that cool when I was in school ,I wouldn't be struggling now. I always knew maths was fun ,but got no assistantce to it from any of my teachers . They were more into marks and stuff rather than learning the purpose of maths

[@LinesThatConnect]

Super cool to see Desmos used for the visuals for a video!

[@MathyJaphy]

Thanks. I'll keep doing videos with Desmos because it's my "thing", but I envy those who deftly use Manim as you have in your SOME1 video. One of these days, maybe I'll learn how to use it.

[@thequixotryworkshop2424]

Very cool! Well explained and nicely animated.

[@simonmoreau4455]

Liked and subscribed juste for the prononciation of Henry, I love the fact that prononciation is important to you Love from France 🇨🇵

[@NStripleseven]

Wow, that’s a really neat way of presenting that.

[@dsanjoy]

That's really awesome. Subscribed. Hope to see more.

[@SprDrumio64]

"Well Rich, what did you think of Van Aubel's Theorem?"

[@Fabsurf101]

it is not only fascinating but there must be some incredible applications waiting to be discovered.

[@mike_the_tutor1166]

Very pretty! Thanks for sharing.

[@destroyer500ful9]

That was very interesting, thanks for the video!

[@rafeef039]

Nice theorem and great presentation Very pretty

[@adityaanand699]

thanks bro for explaining this concept so easily.

[@paisenpaisen]

now this is a great math video, the visualizations are top-notch!

[@mathsangler]

Thank you! A fantastic way to demonstrate a proof!

[@smiley_1000]

Wow, very impressive. When I saw the thumbnail, I did the proof myself with vectors, although my proof was by far less insightful than your construction. Somehow it just all vanishes to zero at the end. The only thing that's really required is that two left rotations form a reflection, which is pretty obvious.

[@maheshbhatt8115]

Thank you for telling about desmo calculator

[@sakettommundrum-teamsolari5693]

Loved it man🔥🔥🔥

[@B34tzepZ]

to answer the last question: really cool

[@omaryahia]

nice, thank you for this illustration

[@Affy420]

Absolutely brilliant

[@yorgle]

Math is freaky. Love it! (And I'm now a new subscriber too!)

[@StewartMcGinnis]

This was fantastic!

[@zaqmko0]

Very nicely done.

[@elysiummaybee]

Damn that is brilliant. I would have never thought of this.

[@patrickstrasser-mikhail6873]

You just contributed to the net Awesome in the universe. Thanks!

[@maximus7947]

This was a very high quality video

[@timothyebbs6443]

Awesome proof, quality video!

[@therealsuper5828]

awesome video man

[@antonoreshkin]

Great video!

[@henk-ottolimburg7947]

How cool? To speak with Balckadders' servant Baldrick: That was pretty cool, m'lord'

[@vcvartak7111]

New maths theorem added to my knowledge

[@hambone4402]

How fun! An elegant proof.

[@ChazFoulstone]

Some math dude discovered this and was like "Nifty".

[@zdikbiodr7341]

Nice proof and very nice graphics indeed. Here is another proof, no graphics. Think of A, B, C, D as complex numbers. Then the midpoint of the side AB of the quadrilateral ABCD is (A+B)/2 and your vector a is (B-A)/2 while a'=i(B-A)/2, i denoting the imaginary unit. The "Start 1" point, thought of as a complex number, equals (A+B+Bi-Ai)/2 and, similarly, the "End 1" is (C+D+Di-Ci)/2. The vector that joins "Start 1" and "End 1" is the difference: v=(C+D-A-B+Di-Ci-Bi+Ai)/2. In the same way the vector that joins "Start 2" and "End 2" is equal to w = (D+A-B-C+Ai-Di-Ci+Bi)/2 (one can get w by cyclic substitution A->B, B->C, C->D, D->A in v). Since iw = v the vectors v and w are equal length and perpendicular. a''=ia'=i^2a=-a so this is consistent with your proof.

[@MathyJaphy]

Yes! Isn't it amazing how complex numbers can make it so much simpler? As I wrote in the description, I actually started with that proof and worked it into a graphical one.

[@zdikbiodr7341]

@@MathyJaphy Dear me, I should have read the description. Thanks!

[@janhornbllhansen4903]

well done

[@LeoStaley]

What software do you use for the animations? Recently 3blue1brown did a sort of challenge/contest where lots of people used his software Manim to make their own educational math videos.

[@MathyJaphy]

I use Desmos Graphing Calculater. There's a link you can follow in the video's description to see the graph that made the video. Yes, I would have entered that contest if I'd had a video ready to go by the deadline. Use of Manim wasn't required. I love what Manim can do and someday I'll learn Python and figure out how to use it.

[@surfer855]

ΑΛΛΗ ΜΙΑ ΕΡΩΤΗΣΗ: ΓΙΑΤΙ SPANE ΧΕΖΕΙΣ ΣΤΟΥΣ ΑΓΡΟΥΣ;

[@aboubakrboubker-qg7sy]

From Morocco..genious..thank you

[@jetaddict420]

You pronounciation of henri van abel makes me cry

[@MathyJaphy]

Alas, me too, now that I've been schooled by several commenters. I added an apology to the description a few days ago.

[@krtm7231]

Here, take my sub. Could you make a tutorial on how you made the desmos demo? Seems very complex and interesting.

[@MathyJaphy]

Interesting idea. I will give it some thought.

[@guessundheit6494]

It makes a change to hear properly edited audio. That was listenable.

[@Sclaraidus]

What if you deform the cube in which the attached squares are on the inside instead of the outside? If the starting shape were a perfect square and the attached squares are on the inside, wouldn’t it just make a single point?

[@JohnSmith-pg3gw]

Fantastic!

[@nicepajuju3900]

Beautiful!!

[@simonking7791]

"A double prime" as you called it is actually called a secont (second in latin, prime is first). Nice video

[@nabeelsherazi8860]

In mathematics the term double prime is the standard

[@simonking7791]

@@nabeelsherazi8860 I am not from an English speaking country and we use secont, but I have watched maths and physics videos in English and they use secont too. Guess it comes down to choice.

[@Anakin_1234]

You were right, this was fun.

[@egohicsum]

Pretty cool! I like

[@varunsohanda2601]

How you eliminated vectors c anad c' from the vector diagram, when they canceled out to zero, was cool.

[@netfaysilec888]

Very cool!

[@paulwestlake4278]

Oddly satisfying to one who has never used more than school math in his life. 😁

[@shragamildiner8472]

This is so cool

[@vijaykumar-kv6gw]

Loved it

[@ПавелГаврилов-я1о]

Super cool!!!

[@yo.pdf1]

No entendí cual es la finalidad de este teorema, que busca demostrar?

[@paracetamol_222]

Wow that's really cool.

[@RichardLightburn]

Very very cool. Next: nine point circle

[@nHans]

I suppose, in this particular case, it is kosher to use vectors or complex numbers to prove a theorem in Euclidean geometry. But in general, I'm always suspicious of chimeric proofs. If you mix-and-match different branches, you have to be absolutely certain about the provenance of the results you're using. Otherwise you run the risk of going around in circles. To use a Graph Theory metaphor, your whole system of axioms, theorems, and proofs need to form a Directed Acyclic Graph (DAG); no loops allowed. True story: - My students have used the distance formula from Cartesian geometry to (trivially) prove Pythagoras Theorem. - They've then used Pythagoras Theorem to prove Euclid's Parallel Postulate / Playfair's Axiom! 😂 Notable exception: If you're solving a Putnam or a Math Olympiad problem, you may use any "well-known" result from any branch of math.

[@LJL0619]

Nice video. But please drop the background music. What purpose does it serve, except to distract from your voice?

[@MathyJaphy]

Another vote against. Got it. I think it’s subliminally soothing and would add a touch of class if it were classier music. Besides, a distraction from my voice might be considered a plus. :-)

[@Kengur8]

Does it work with triangles variant where you need to prove equilateral?

[@MathyJaphy]

You’re referring to Napoleon’s Theorem. It might be possible to prove it using vectors like this, but I suspect it wouldn’t be as visually obvious, and might require some non-trivial math. Definitely something to think about.

[@MathyJaphy]

Thanks for the inspiration! kzbin.info/www/bejne/lXXTeZKAirVjm7s

[@panzielynsky2922]

Well its actually fun thing to learn. Its a shame i never heard about it in school

[@ZephenTheGreat42]

4:55 ummm did you accidentally prove how to perfectly find the midpoint of a line on any triangle?

[@MathyJaphy]

Ha! No, since you have to have the midpoint to begin with in order to construct the small squares. But perhaps I proved that if you construct squares on the two equal sides of an isosceles triangle, and on each of the two halves of the third side, then the lines connecting opposite centers of the squares will cross at the midpoint of the third side.

[@d.e.p.-j.7106]

Does it generalize somehow when the four lines aren't in a plane?

[@dhrubajyotidaityari9240]

Beautiful 😀😀

[@GrandMiserigueux]

There a lot of shortcuts in this proof. Ex: why a'+b'+c'+d'=0?

[@kylesmith7086]

This tool would have been nice to have in the 90s.

[@bayupatten4777]

this was definately cool

[@carlos6126]

it was pretty cool dude

[@beeruawana6662]

Vety good question

[@BeaDSM]

Super cute; thanks!

[@attila3028]

amazing

[@aryakadam7892]

Yes really cool 👍

[@oceanbeans8546]

The way he said henri van aubel is hilarious for me as a Dutch person

[@MathyJaphy]

Apologies! I should have done more research. :-(

[@oceanbeans8546]

@@MathyJaphy no worries! I’ve never heard a non dutch person say dutch things correctly. It’s just funny not offensive or anything

[@chittasticchitta1164]

This is trippy

[@deserado11]

... luv it ...

[@adickurig]

Are there any applications for this though?