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  • We'll talk about one here, um, that I learned when I was in high school and the seemingly miraculous methods to solve it, which is my favorite thing from the training for the International Math Olympiad back in Bulgaria.

  • 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?