Name: 奇跡の証明（プトレマイオスの定理） - Numberphile (A Miraculous Proof (Ptolemy's Theorem) - Numberphile)
Uploaded: 2021-01-14T10:21:42.000Z
Duration: 38 min 28 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

様態の副詞

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.

基礎受動態

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?

Absolutely well, What do you know about it?

A squared plus B squared is equal to C squared.

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

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

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

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

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.

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.

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?

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

Also happened to be called in more over their intersection.

Oh, is it equal distances from each Vertex s?

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

So now let us let us backtrack a little bit.

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.

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?

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.

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?

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.

Inversion is obviously some transformation of the plane.

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?

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

How about this one That's called rotation.

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.

Will the elephant look after each of those transformations?

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

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奇跡の証明（プトレマイオスの定理） - Numberphile (A Miraculous Proof (Ptolemy's Theorem) - Numberphile)

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