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• What do Euclid,

• twelve-year-old Einstein,

• and American President James Garfield have in common?

• They all came up with elegant proofs for the famous Pythagorean theorem,

• the rule that says for a right triangle,

• the square of one side plus the square of the other side

• is equal to the square of the hypotenuse.

• In other words, a²+b²=c².

• This statement is one of the most fundamental rules of geometry,

• and the basis for practical applications,

• like constructing stable buildings and triangulating GPS coordinates.

• The theorem is named for Pythagoras,

• a Greek philosopher and mathematician in the 6th century B.C.,

• but it was known more than a thousand years earlier.

• A Babylonian tablet from around 1800 B.C. lists 15 sets of numbers

• that satisfy the theorem.

• Some historians speculate that Ancient Egyptian surveyors

• used one such set of numbers, 3, 4, 5, to make square corners.

• The theory is that surveyors could stretch a knotted rope with twelve equal segments

• to form a triangle with sides of length 3, 4 and 5.

• According to the converse of the Pythagorean theorem,

• that has to make a right triangle,

• and, therefore, a square corner.

• And the earliest known Indian mathematical texts

• written between 800 and 600 B.C.

• state that a rope stretched across the diagonal of a square

• produces a square twice as large as the original one.

• That relationship can be derived from the Pythagorean theorem.

• But how do we know that the theorem is true

• for every right triangle on a flat surface,

• not just the ones these mathematicians and surveyors knew about?

• Because we can prove it.

• Proofs use existing mathematical rules and logic

• to demonstrate that a theorem must hold true all the time.

• One classic proof often attributed to Pythagoras himself

• uses a strategy called proof by rearrangement.

• Take four identical right triangles with side lengths a and b

• and hypotenuse length c.

• Arrange them so that their hypotenuses form a tilted square.

• The area of that square is c².

• Now rearrange the triangles into two rectangles,

• leaving smaller squares on either side.

• The areas of those squares areand b².

• Here's the key.

• The total area of the figure didn't change,

• and the areas of the triangles didn't change.

• So the empty space in one, c²

• must be equal to the empty space in the other,

• a² + b².

• Another proof comes from a fellow Greek mathematician Euclid

• and was also stumbled upon almost 2,000 years later

• by twelve-year-old Einstein.

• This proof divides one right triangle into two others

• and uses the principle that if the corresponding angles of two triangles are the same,

• the ratio of their sides is the same, too.

• So for these three similar triangles,

• you can write these expressions for their sides.

• Next, rearrange the terms.

• And finally, add the two equations together and simplify to get

• ab²+ac²=bc²,

• or a²+b²=c².

• Here's one that uses tessellation,

• a repeating geometric pattern for a more visual proof.

• Can you see how it works?

• Pause the video if you'd like some time to think about it.

• The dark gray square is

• and the light gray one is b².

• The one outlined in blue is c².

• Each blue outlined square contains the pieces of exactly one dark

• and one light gray square,

• proving the Pythagorean theorem again.

• And if you'd really like to convince yourself,

• you could build a turntable with three square boxes of equal depth

• connected to each other around a right triangle.

• If you fill the largest square with water and spin the turntable,

• the water from the large square will perfectly fill the two smaller ones.

• The Pythagorean theorem has more than 350 proofs, and counting,

• ranging from brilliant to obscure.

• Did you enjoy this lesson?

• If so, please consider supporting our non-profit mission by visiting PATREON.COM/TEDED

What do Euclid,

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# 【TED-Ed】How many ways are there to prove the Pythagorean theorem? - Betty Fei

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Anita Lin posted on 2017/09/12
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