I'm assuming that you just watched the Illuminati confirmed Video and that you're a hard core
mathematician and that you now want the proof that Morley's theorem really works.
I'm going to give you a proof now and it's going to be an absolutely amazing
proof. Now Morley's theorem is notoriously hard to show. It's very
messy, 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 I'm going to try and make it even more
accessible by animating it the way I do it here on Matholger. So it's my tribute to John
Conway but also it's just an absolutely beautiful thing so one should
have a proof somewhere. Just in case you haven't watched the Illuminati confirmed video maybe just go there
and watch it but just a quick recap what happens is you
take an arbitrary triangle, you trisect the angles, cut them into
three equal pieces and then you extend these cuts and they meet in three points
and the three points always form an equilateral triangle which is something absolutely
miraculous. It is called Morley's miracle and here's a proof, here 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. And the next thing we need to remind
ourselves of is that six times sixty is 360, right. So you can cut a full circle into
six equal parts of sixty degrees. Next thing is if you've got two triangles and they
share two angles they are similar which means that one can be scaled up to the
other. Then if two triangles actually share two angles plus the segment between the
two angles, then they are congruent which means means you can take one and superimpose it onto
the other one you know so they're really the same triangle. So let's start the proof. The first
thing you have to realize is that this angle plus that angle plus that angle is
180 degrees so if we add up the little angles, the thirds like up there
we've got the thirds all put together they add up to 180 divided by three which is sixty
degrees. Just keep that in mind whenever you see red, green and blue
together you see a 60 degree angle. Alright now
Conway's ideas is to kind of go backwards with his proof. So he says, so we've got this
configuration here. Let's just assume this works out, it's really an
equilateral triangle right in the middle. What are the angles that we see here?
So, for example, what's that angle here?
Well we see a blue we see a red if we add a green we've got sixty degrees and then we need another
two sixties to make 180. So there is really no choice, it's gotta be that. And same sort of
thing happens for this and that. So here we've got these sort of angles happen.
There's no choice it's gonna 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.
And we can go around this point here and add up things there.
And so what do we get: b plus c plus three times sixty plus green is ... 360.
And there we've got another one, another equation and another equation and another
equation. 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. You
can actually choose one of those angles any way you want and you can satisfy
those six equations. That's not quite good enough but what we can do, of course, is
we can have a really close look at some specific examples of these triangles and then
come up with a conjecture as to what these angles might be and the conjecture is
actually that they are as nice as possible. So our conjecture is that they are as nice
as possible, like that.
This is actually going to be part of what we want to prove. So what we want
to prove is that the trisetions give this equilateral triangle in the middle but in
addition to this we also want to prove that the angles are as nice as this. So why
are they really nice. Well have a look at this. In this triangle, for example, we've got red,
green, blue, that's 60 degrees plus two 60s that's going to be 180.
If you go around this corner here, we've got 1, 2, 3, 4, 5 sixties plus another 60 blue, green, red,
and this kind of goes all the way around so the angles add up the way
it's supposed to be but in the best possible way.
We want to show that everything's best possible so the
triangle that is formed in the middle is equilateral plus this nice
distribution of angles all the way around. It is suggested by this kind of
set-up what we are going to try and do. We're actually going to try and go backwards. So we're
going to start with just this triangle here in the middle and now
we're going to build up the rest and see whether we can actually get things to
fit together like this properly. Okay, so the first thing we do is we know, well, what
the angles here are supposed to be. So we're going to start with this and that gives
us this triangle here and same thing around here and same thing there.
We are going to start like this, kind of try to reconstruct that triangle that we just saw
reconstruct and prove at the same time that it actually fit together like this.
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 blue. And once we've shown that things already look pretty good.
Then we have to do this three times. So we have to show that this angle is blue and this is green.
And then one more time, green and red. And, actually, the argument is exactly the same for this part and that part as
it is for the first part. Once we finished
showing that this is necessarily red and this is necessarily
blue with our specific choice of angles we are done.
And now we go on fast forward, heavy duty math, but it's really just very pretty.
We take this triangle here and make a copy of it, flip it,
attach it. Now we take that point and pull it out until that angle turns into a blue angle.
So, like that. Now we've got that big triangle here and we just ask what's that
big angle there. Well it's again blue, red, if it had a green and two 60s then it's going
to be okay. So this is forced, this angle is forced to be what we
expect it to be, just like before.
Okay, now what about that angle here.
Well if you already look at 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 and sixty degrees. So that is all fine.
So what have we got now?
Well we've got two angles that are the same plus these two sides are the same.
So this triangle here fits nicely into this corner and the only thing we have
to convince ourselves of now is that actually these two sides are also the same
and then we're ok. (Giuseppe) They could be different lengths. Yes, they could be different lengths.
That's the only thing that can go wrong, so we have to really figure out that
they are the same length. Okay, so let's go back to this flipped and attached triangle.
Make a copy of the heavily drawn segment, insert it in a second way. Now what
we've got here is an isosceles triangle which means that these two sides are the same but that
also means that this angle and that angle are the same.
So this angle is also this guy.
Now, we highlight something else, we highlight this triangle there.
Now we can figure out what that angle is and it's clear that it
has to be that one, you know, pretty obvious. And so what that means now is that if we compare
this triangle to that triangle we get one angle in common, a second angle
in common, plus the side in the middle in common and that means that the two
triangles are congruent and that also means that this side is the same as
that side. Tada. And that means that this guy here fits in just like that. (Giuseppe) Beautiful. And basically
you've won here at this point in time because you now just repeat. You make a copy of this
flip, attach, extend. Run the same argument again. That means that this guy
fits in. Run the same thing again and you are done. So, you've reconstructed this whole thing
and it all fits together and so everything works. Probably the best proof of Morley's miracle in
existence. (Giuseppe) By John Conwey. By John Conway, and animated by the Mathologer. (Giuseppe) I am starting to create triangles with all these angles, so should get some pizza.
Yes, salmon pizza. And this is Morley's miracle confirmed.