# Math is Illuminati confirmed (PART 2): Morley’s Miracle

I'm assuming that you you just watched the Illuminati confirmed video and that you're a hardcore mathematician and that you now want to prove that your model is theorem really works so I'm going to give you the proof now and it's going to

Be an absolutely amazing proof now this Morley's theorem is notoriously hard to show it's it's very messy it used to be very messy until recently when my favorite mathematician in the whole wide world John Conway from

Princeton University came up with a very very beautiful accessible proof and so I'm going to try and and make it even more accessible animating it the way I do it here on mythology okay so it's my tribute to John Conway but also it's

It's just absolutely beautiful thing so one should have a proof somewhere well just in case you haven't watched the Illuminati confirmed video maybe just go then and watch it but you know just a quick recap what happens is you take an

Arbitrary triangle you try sector angles and we kind of cut them into three equal pieces and then you extend these cuts and then meet and three points and the three points always form an equilateral triangle which is something absolutely

Miraculous it's called Molly's miracle and here's a proof is that amazing proof okay so the only thing we really need here is that the sum of the angles in a triangle is 180 degrees everybody knows this next thing we need to you know kind

Of remind ourselves that 60 times 60 is 360 all right so you can't cut a full circle into six equal parts 60 degrees next thing is if you've got two triangles and they share two angles then they're similar which means that one can

Be scaled up to the other okay and then if two triangles actually share to two angles plus the side in between the two angles then they actually congruent which means you can take one and put it on to the other one you know so they're

Really the same triangle so let's start the proof so the first thing we have to realize is that well this angle Plus that angle Plus that angle add up to 180 degrees so if we add up the little angle theater the thirds you know

Up there what the Third's kind of fall together they add up to 180 divided by three which is sixty degrees right so sixty degrees okay just keep that in mind whenever you see red green and blue together you see a 60 degree angle okay

All right now Conway's idea is to kind of go backwards with his proof so he says mmm so we've got this configuration here let's just assume this works out it's really a sixty

Degree an equilateral triangle right in the middle what are the angles that we see here so for example what's that angle here okay well we see a blue we see a red if we add a green we've got 60 degrees and then we need another two 60s

To make 180 so there's really no choice it's got to be that right and the same sort of thing happens here for this and that so here we've got these sorts of angle happen there's no choice here it's got to be like this now there's more

Angles here six more angles and we've got a couple of clues as to what these angles might be so we can go around this triangle there and say well a plus B plus red should be 180 right and we can go around this this point here and add

Up things there and so what do we get C plus B plus three times 60 Plus greenness 360 that's right and they've got another one another equation and another equation another equation and another equation so we've got six

Unknowns here and six equations and usually when you've got something like this you can actually solve you know except for this one you can't it's it's almost determined but not quite so I mean if you if you can actually choose

One of those angles and it's anywhere you want and you can you can satisfy those through six equations that's not quite good enough but what we can do of course we can have a really really close look at some specific examples of these

And then maybe come up as a conjecture as to what these angles might be and the conjecture is actually that there are as nice as possible so our conjecture is that they're as nice as possible like that okay so this is actually going to

Be part of what we want to prove so what we want to prove is that the Tri sections give this equilateral triangle in the middle but in addition to this we also want to prove that the ink as nice as this so why are they really

Nice well just look at this so this triangle here for example I've got red green blue let 60 degrees plus two 60s that's going to be a hundred eighty right if you go around this corner here we've got one two three four five six TS

Plus another 60 blue green red and it kind of goes all the way around so the angles kind of add up all all the way it's supposed to be but in a just the best possible way about in the best possible way so we want to show that

Everything's best possible right so the triangle is formed in the middle is equilateral plus this nice distribution of angles all the way around and well it's suggested by this kind of setup what we're trying to try and do actually

Going to try and go backwards right so we're gonna start with just this this triangle here in the middle and now we're going to build up the rest and see whether we can actually get things fit together like this properly okay so

First thing we do is we know well what the angles here is supposed to be so we're gonna start with this and that gives us this triangle here and same thing around here and same thing there so we're gonna just start like this okay

Kind of try to reconstruct that that triangle that we just saw reconstruct it and prove at the same time that it actually fits together like this yeah okay then we draw a line here and now what we really have to show is that this

Angle is red and that this angle is and once I've shown that it looks already pretty good right then we just have to do it as three times so we have to show that this inglis blue end is and then one more time right green and red and

Actually the argument is exactly the same for for this part and that part as it is for the first part so once we've finished you know this showing that this is necessarily red and this is necessarily

Blue with our specific choice of angles we are done really okay so here it goes and now we go on fast-forward heavy-duty Matic it but it's it's really just very pretty okay so we take this triangle here big copy okay so make a

Copy flip it attach it I would take that pointy and pull it out until that angle turns into blue angle so like that okay now we've got this big big triangle here and we just ask what's that angle there well it's again blue red if it added

Green and 260s then it's going to be okay so this is forced right this angle is forced to be what we expect it to be okay just like before okay now what about that angle here well if you already look

At what what's here it's pretty much what we want it to be so it's also clear that this one has to be red end and sixty degrees so that that's all fine you can okay so what have you got now well we've got two

Angles that are the same plus these two sides are the same so this triangle here fits in nicely into this corner but the only thing that we have to convince ourselves now is that actually these two sides are also the same than it then

Then then we're okay but there could be different lengths there could be different lengths that's the only thing that can go wrong so we have to really figure out that they're the same lines again so okay so let's go back to this

Flip and attach triangle there make a copy of the heavily drawn segment insert it in a second way and now what we've got here is a is a sauceless triangle and it means these two sides are the same what it also means that this angle

And that angle are the same right so this angle is also this guy here okay now we highlight something else we highlight this triangle there and so we've got an angle there and now we can figure out what that one is

And it's clear it has to be that one you know pretty obvious right and so what that means now is that if we compare this triangle to that triangle we get one angle in common the second angle in common plus the side in the middle in

Common and that means that the two triangles are congruent that's a congruent and that also means that this it is the same as that side tada and that means that this guy here fits

In just like that beautiful and that's basically you've won here at this one time because now it just repeat right you take a copy of this flip attach extend sit run the same argument again it means that this guy

Fits in right you run the same thing again and you're done right so you've reconstructed this this whole thing and it all fits together and so everything works for probably the best proof of Morris miracle in existence by John

Conway by John Conway legend Kombi and animated by their mythology oh yes yes Salman pizza okay and this is Molly's miracle confirm