Three pretty geometric theorems, proved by complex numbers

Bottema'a theorem, van Aubel's theorem and Napoleon's theorem, are all brilliant. If you have never seen them before, you will be amazed! I prove then all by working in the complex plan, as van Aubel did in 1878.

[@peterhall6656]

A ripsnorter. I knew Fourier did bureaucratic jobs for Napoleon but wasn't aware that Laplace had taught him. Imagine the dinner scene over a pretentious Bordeaux: Laplace: Emperor, may we move on from world domination for a moment to this cheeky little theorem? Napoleon: How is your Russian?

[@jimmathy]

Thanks! I'd forgotten the connection to Fourier, thanks for reminding me.

[@lugiagaurdien773]

Absolutely gorgeous video. We need more of this on RUclips.

[@gregorycampbell5142]

One of my favorite things about complex numbers is the surprising degree to which they:
-simplify complexity in 'Real' situations (not to mention geometrical ones, amongst many others);
-can be applied to, seemingly, ANY type of analysis, usually (probably always) in meaningful, useful ways;
-can inform and support one's intuition, despite being rooted in the wholly unintuitive concept of negative even roots.
We should have called them Ironic Numbers ... except then they wouldn't be as ironic ... which, uh, is actually kind of ironic? Hmm... 🤔

[@wasylbasyl]

"despite being rooted" nice one

[@JasonCunliffe]

Inspired Numbers

[@bowlseriw]

The unreasonable effectiveness of complex numbers

[@wickedpawn5437]

I believe the last theorem is where Symmetrical Components in Electrical Engineering come from. Representing a 3-phase unbalanced system by the superposition of three balanced systems.

[@jimmathy]

Wow, that's intrguing! However, I don't know anything about Elecrical Engineering, so I can't comment.

[@twanvanderschoot9667]

Nice demonstration. But in case of Napoleon's Theorem, it is sufficient to prove that the lengths of the side of the triangle are equal, you don't need to introduce rotation over 120º.

[@tunguyenhoc341]

Beautiful. Brilliant. So enjoyable lesson. Thanks.

[@jimmathy]

Thank you.

[@renesperb]

Very nice geometric problems . It is also good to see ,especially for people who didn't like complex numbers , how much you can achieve
by a clever application of them.

[@alikaperdue]

I used complex numbers miners to find all the 3rd points of similar triangles with two fixed points. It was a Riemann sphere.

[@niksforeve2805]

Indeed, beautiful!

[@jimmathy]

Many thanks!

[@justsomeboyprobablydressed9579]

Very enjoyable video. Thank you.

[@a00b00c]

Thank you! It is pretty simple. I thought it would have been harder to understand.

[@cauchym9883]

Fantastic video!

[@aurelienperdriaud108]

Beautiful!

[@gamespotlive3673]

Brilliant man, this is a great video.

[@GregShyBoy]

For the last part of the third theorem, there's a way to avoid "knowing" the cubic root of 1. But it'll require calculating a module of a complex number. Calculations may seem longer but (arguably) less knowledge is required.
Overall - awesome job! This lesson totally answers the scholar's question "Who even needs these imaginary numbers?"

[@jimmathy]

Thanks!

[@jursamaj]

But it's quite easy to derive the needed value anyway, without long calculations, just from symmetry & pythagoras.

[@antoniorose2461]

Amazing!

[@mohammadjaveed7404]

Beautiful explanation Thanks

[@physicsman1270]

love the napoleon bit

[@pauselab5569]

I used to prove random common Euclidean geometry problems with complex coordinates. Works much better than real coordinates most of the time though not always better than using theorems

[@040_faraz9]

Totally love it. Can you recommend any book for geometrical theorems like these?

[@tweytwan3890]

Geometry in Figures by Arseniy Akopyan

[@tinafeyalien]

I don't get the 1/sqrt(3) bit. Half the height of an equilateral triangle of side length 2(y-x) is [sqrt(3)/2]*(y-x)

[@Risu0chan]

yes but the center is not at half the height, but at one third of it.

[@mikesteele5935]

Marvelous and memorable.

[@jimmathy]

Thanks

[@kailashvardhan7085]

Are you the man who solved the Market. If yes please teach me sir. It's needed.

[@tissuepaper9962]

My opinion of this video can be summed up by the following: complex numbers fucking rock! Repurposing algebraic machinery to perform geometric calculations, the elegance of it is astounding.

[@Dr.1.]

😮😮😮 beautiful theroems

[@diskgrinder]

What’s the Mac app you’re using?

[@jimmathy]

Geogebra

[@landsgevaer]

Some comment should mention the Petr-Douglas-Neumann theorem in this context. Allow me to.
I knew about the theorem but had forgotten its name. Thanks for triggering me to look it up again. Wonderful little tidbit that!

[@jimmathy]

Yes, that's a great theorem.

[@landsgevaer]

@@jimmathy And not a week later another video comes out ( ruclips.net/video/WLAW5yz5O3E/видео.html ) on the same. Nice combo of coincident videos!

[@retrothink]

What software are you using?

[@jimmathy]

geogebra

[@parcton9716]

Nice video!

[@avyakthaachar2.718]

So cool ❤

[@arifroktim3366]

What software are you using for the demonstration?

[@jimmathy]

I use geogebra - give it a try, it's marvelous!

[@mazxki]

What was the software used in this video?

[@jimmathy]

I used geogebra.

[@dvdortiz9031]

Waterloo

[@mankalememartin1371]

brilliant!
now how can one go about proving concurrency of lines using complex numbers?

[@wesleydeng71]

RIP Jim.

[@jimmathy]

I'm not that Jim Simons!

[@abogadojon]

Paraphrasing Mark Twain, reports of your demise are over-emphasized, correct? Good to hear it, Mr. Jim.

[@nathansmith2168]

Definitely cool video but I think it is important to point out that complex numbers are in no way necessary for these proofs, and this isn't even really justification for the utility of them, since I'm pretty sure the proofs could just be phrased in terms of the basic rotation formulas and vectors. The complex numbers just provide a convenient and natural shorthand.