The theory is called Qala Misty Room and the method.

Well, wait a few minutes before we get there.

Well, let's start with something that everyone knows.

What is the fear and braided that you think everyone knows geometry?

Pythagoras.

Absolutely well, What do you know about it?

Even I know this one.

A squared plus B squared is equal to C squared.

The old people know this theorem but didn't know how to prove it.

Soap Ptolemy's theory.

M iss somehow related to this one.

So who do you think it's strong up?

Ptolemy by Thackery Stronger, Stronger They're both Greeks.

Well, Pythagoras is more famous.

That is true, but they have to compete with each other and we shall see.

I need another brown sheet already.

Yeah, so we're going to draw a large circle.

We need four points on this circle.

They form a so called sickly quadrilateral.

It just means they lie on the circle and you can also draw the The AG knows so what Ptolemy tell us about these six segments.

It's very beautiful.

It says If you multiply the opposite sides and out of those products, you will get the product of the two diagonals.

So a B times C ndy plus a D times B C must be quo to the product of the two diagonals A.

C times beauty.

And this is what columnists theorem tells us and it doesn't matter where those points on the circle as long as they in this order, it's gonna work.

So this is a property about points on a Cirque.

Now let's see.

I said that Pythagoras in Ptolemy's are related course stronger if I can prove Pythagoras knowing.

Atala, Mysterion, the dollar Mrs Stronger.

Okay, so it would be like a parent or yeah, soap.

Ptolemy would be a parent off.

Put Sanders.

Is this possible?

Certainly.

So if I draw a right triangle and then take exactly the same triangle and flip it over, what kind of a shape do you think?

Brady?

I'm going to get a rectangle.

It is precisely rectangle.

So we have a BCD for the Vergis is now a rectangle is a very symmetric figure.

It hears All right, ankle So?

So it's to the other knows.

Also happened to be called in more over their intersection.

Oh, is it equal distances from each Vertex s?

So they must be on a circle.

Those points exactly.

But this soon as we have four points on the circle guess what happens.

Ptolemy, you say?

Yes, but Ptolemy would kick in.

So now let us let us backtrack a little bit.

We started with a right triangle.

Let us name its sides A B and the diagonal will be C.

Okay, but then this side should be a And this side should be being the other diagonal should also be C.

Let's hit this picture with columnists.

So what do we have?

The product of the opposite sides.

A times it, plus the product of the other opposite sides.

Be times be must be equal to the product of the today are gonna see time.

See, Does this look like a feeling we've seen before?

It looks good.

That is precisely the piss Agree in theory, which means that Ptolemy is stronger.

Pythagorean theorem is a special case of column Mystere, but Ptolemy's theory and can do a lot more, and so we create a new method today.

Actually, I've known about this method for a long time that will solve columnist here.

So actually Ptolemy's can be proven in many different ways.

If you go, you will see lots of different proofs.

You will see a proof in plane geometry, which is beautiful but very tricky toe come up with.

You can see technical proofs using trigonometry who likes cows or using even complex numbers.

Very sophisticated.

But none of these approaches really shows the true elegance and simplicity off Stalinist era.

Um, none of them shows you why it is really true.

The new method that we will talk about today, called inversion in the plane, looks originally like a mysterious black box because it will turn our circle and pol amiss situation into something that 1/3 grader can tackle.

So it will turn this whole picture into something there receivable on Lee.

Three points that the lined up a one b one and see.

What could 1/3 grader tell me about this picture?

What is true?

Brady.

I am Bay and then B and C makes a whole lot correct.

If I add the two shorter once, I must get the longer one look back, a dollar Mystere and we're almost doing something similar where having two things on getting 1/3 1 Indeed, this be the same situation, which in a different world in an inverted world in the answer is yes.

So what is inversion?

Inversion is obviously some transformation of the plane.

It's same point.

Some points to some other points.

So let's think about other transformations in the plane that we all know.

So what would be one the two Brady have heard about, like reflections and things like that?

Is that what we're doing?

Correct.

Yeah, you can reflect across the line.

Now that's that's a perfectly good transformation.

How about this one That's called rotation.

About a point, definitely.

How about this one?

That's called translation, and there's another one.

When you click on a map to expand it to zoom it out, zoom in.

That's what's called a re scaling or dilation.

So suppose I draw on elephant?

What do you think?

Will the elephant look after each of those transformations?

But it looked like a rabbit or would it look like an elephant?

It will.

It will keep its elephant nous correct.

It may kind of flip flop, but it will still pretty much like an elephant.

Well, our inversion is a completely different and because we saw how it transform eight's a circular picture into Alina one.

So it must do something horrible to shapes.

If you take your nice elephant and hit it with that inversion, what will turn out in the end?

It's something that might look like a lot in shoe from the movie.

Completely ridiculous.

So we need to have control over the situation.

What is going on?

Cellphone inversion.

You need the circle into the boss.

You can draw the circle anywhere in the plane like and as big as you like and we mark it center.

Oh, let's say we take a point outside of this circle X.

Where will this point go?

It's like a small algorithm here to start to the point X, we connected with the center.

We draw attention through X until it hits the circle at T and then we drop a perpendicular toe All ex.

So our point X if it is outside of the circle, we'll travel into the circle in tow.

Extra.

This actually can happen on any point in the plane.

For instance, if I take something very close here and we just creepy we connected toe all we draw changing and we drop a perpendicular.

That will be where Why one will go If I take any point outside the circle, Where do you think it will go?

It looks like we'll go inside the correct That can happen to any point outside the circle.

Now we have more points to worry about.

What happens if you start with a point inside the circle?

Well, we want to be consistent.

So the logical thing is to reverse this algorithm.

If I take a point z Well, I still need the race through Z and all because everything important looks like gonna lie there.

But then I have to backtrack and erect a perpendicular at Z to this ray, teaching the circle and finally drawing my change into the circle until it in their sex.

Our original array in point Z one.

So if you start inside the circle, you go outside of the circle.

That's what inversion does to you.

okay.

Mmm.

Mmm mmm.

I think we're still missing a few points.

Which points?

Did they not talk about the ones on the circle?

Correct.

So suppose I take a point about you.

So you is on the circle.

I want to treat it as if it is a point outside of the circle.

So we're going to apply our first algorithm.

We connected toe.

Oh, next you remember.

I'm supposed to draw attain JJ into the circle through that point.

Okay, well, the point is already on the circle, so it's just kind of kind of easy.

And the changes willing There six the circle in our next point.

But that's going to you already.

Okay.

And next, they have to drop a perpendicular to my line until it intersex that ray.

Okay, I'm gonna drop a perpendicular to this line.

But I'm already there, so I literally have not moved from you.

So any point on the circle is going to go into itself.

And that kind of makes sense.

You will take outside of the plane, this infinite plane, and you will squash it inside.

It will take the inside.

You will sprayed outside.

Sprayed it all over, but the points on the board, the line I'm gonna stay fixed and not gonna move anywhere.

What happens to the center?

The center looks like it's an inside point.

So we should apply our second album.

But can it possibly work?

That center is a very strange point.

So the first thing we have to dio by our second algorithm, if you remember Point Z, we need to connect it to the center.

Well, how do we connect it to the center?

I saw you one point.

How do you draw that, right?

Yeah.

I don't have a ray here, so we are already kind of stuck with our algorithm, But maybe you'll say it doesn't matter which way we just draw around the one.

Okay.

Very good.

No problem.

Next I'm supposed to erect a perpendicular.

Certainly.

No problem.

I'm supposed to draw attention to the circle until aw, we are in trouble.

They parallel.

There was no way these two lines of running So you may say Oh, you go somewhere it infinity.

Okay, in that direction.

But what happens if I chose a different direction?

Is gonna go in that direction.

It infinity for our purposes off proving Ptolemy's theory.

Um, we don't need to even bother with the center of inversion.

We will not define inversion at the center and you're asking all that.

That doesn't seem right.

It should be defined everywhere.

I beg your pardon?

If you, for example, I know a thing about functions.

One of the first function that would look like this is f of X equals to one over X Brady.

Can I plug in any number here?

I think we're in trouble.

Campos era.

We can't zero there, so this function is not defined.

It zero.

But it's a perfectly good function and you can see it all over the place.

So why not leave inversion not defined in the scent?

Now, when I was in middle school, they used to tease us with the following question.

How can you catch all lions in Africa?

All right, will build the fence a really strong fans around the circle.

Where would you like to stay?

Brady Outside the circle inside the circle.

I want to be free on the outside.

Oh, that's where the lions are.

So you're gonna make sure that inside the circle there no lions I thought I was alive.

Oh, well, okay.

I wanna be I wanna be away from the line.

So you stay inside this lion free circle, and then you hit the world within version.

So all the lions outside don't come in, but you will be safely outside and observing them.

Well, I don't know if that's a worthy goal, but you see the excitement of fifth graders, how much they could foresee the power of inversion.

What are the basic properties of inverse?

We already saw a few of them points outside the circle, going points inside.

Go.

But do they do this randomly?

What happens if I apply inversion twice in a row?

So point X from the outside is gonna go into point X one on the inside?

But if I do this once again, I'll simply reverse my algorithm and go back to X.

So what happens is that points in the plane are sort of Marat and they flip flop from inside and outside.