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What do Euclid, twelve-year-old Einstein, and American President James Garfield have in common?
欧几里得，十二岁的爱因斯坦，以及美国总统詹姆斯·加菲尔德，他们有什么共同点？
They all came up with elegant proofs for the famous Pythagorean theorem, the rule that says for a right triangle,
他们都对毕达哥拉斯定理（勾股定理）做出了精彩的证明，这个定理是说，对于一个直角三角形，
the square of one side plus the square of the other side is equal to the square of the hypotenuse.
一边的平方加上另一边的平方，等于斜边的平方。
In other words, a2+b2=c2.
换句话说，a2+b2=c2。
This statement is one of the most fundamental rules of geometry,
这是几何学中最基本的定理之一，
and the basis for practical applications, like constructing stable buildings and triangulating GPS coordinates.
也是实际应用的基础，比如建造稳定的建筑，或对GPS点进行三角测量。
The theorem is named for Pythagoras, a Greek philosopher and mathematician in the 6th century B.C.,
这个定理以毕达哥拉斯命名，他是公元前6世纪的希腊哲学家和数学家，
but it was known more than a thousand years earlier.
但是该定理在此之前的1000多年就出现了。
A Babylonian tablet from around 1800 B.C. lists 15 sets of numbers that satisfy the theorem.
公元前1800年的巴比伦石板上列出了满足该定理的15组数字。
Some historians speculate that Ancient Egyptian surveyors used one such set of numbers, 3, 4, 5, to make square corners.
一些历史学家认为，古埃及勘测员利用譬如3，4，5的数组，来形成直角。
The theory is that surveyors could stretch a knotted rope with twelve equal segments
该理论认为勘测员可以伸展一个被绳结分成12份的绳子，
to form a triangle with sides of length 3, 4 and 5.
来形成边长为3，4，5的三角形。
According to the converse of the Pythagorean theorem, that has to make a right triangle, and, therefore, a square corner.
根据毕达哥拉斯的逆定理，这就可以形成一个直角三角形，因此，便可形成直角。
And the earliest known Indian mathematical texts written between 800 and 600 B.C.
已知最早的印度数学记录出现在公元前800至600年间，
state that a rope stretched across the diagonal of a square produces a square twice as large as the original one.
其说明穿过正方形对角线的绳子，可以产生比原来正方形面积大一倍的正方形。
That relationship can be derived from the Pythagorean theorem.
这种关系源于毕达哥拉斯定理。
But how do we know that the theorem is true for every right triangle on a flat surface,
但是我们怎么知道这个定理对平面上的每个直角三角形都成立，
not just the ones these mathematicians and surveyors knew about? Because we can prove it.
而不是一些数学家和勘测员所推测的呢？因为我们可以证明它。
Proofs use existing mathematical rules and logic to demonstrate that a theorem must hold true all the time.
利用现有的数学定理和逻辑，我们可以证明该定理总是成立。
One classic proof often attributed to Pythagoras himself uses a strategy called proof by rearrangement.
经典证明是毕达哥拉斯自己做出的，他利用了一种名叫排列的证明方法。
Take four identical right triangles with side lengths a and b and hypotenuse length c.
取四个全等的直角三角形，两边分别长a和b，斜边长c。
Arrange them so that their hypotenuses form a tilted square. The area of that square is c2.
将它们排列，使它们的斜边形成一个正方形。这个正方形的面积是c2。
Now rearrange the triangles into two rectangles, leaving smaller squares on either side.
现在，重新将三角形排列成两个长方形，让各边形成一个小的正方形。
The areas of those squares are a2 and b2. Here's the key.
这些正方形的面积分别为a2和b2。这就是关键。
The total area of the figure didn't change, and the areas of the triangles didn't change.
图形的总面积没有改变，三角形的面积没有改变。
So the empty space in one, c2 must be equal to the empty space in the other, a2 + b2.
所以第一幅图中的空白部分，c2，必须等于另一幅图中的空白部分，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.
另一种证明来自希腊数学家欧几里得，这种证明也被2000年后12岁的爱因斯坦提出。
This proof divides one right triangle into two others and uses the principle that
这种证明将一个直角三角形分为两个部分，利用了如下定理，
if the corresponding angles of two triangles are the same, the ratio of their sides is the same, too.
如果两个三角形对应的角相同，那么它们的边的比例也是相同的。
So for these three similar triangles, you can write these expressions for their sides.
所以对这三个相似三角形，你可以写出它们的边的表达式。
Next, rearrange the terms. And finally, add the two equations together and simplify to get ab2+ac2=bc2, or a2+b2=c2.
下一步，整理各项。最后，将两式相加，化简得到ab2+ac2=bc2或a2+b2=c2。
Here's one that uses tessellation, a repeating geometric pattern for a more visual proof.
还有一种用了曲面细分法，这是一种重复几何图案的更加视觉化的证明。
Can you see how it works? Pause the video if you'd like some time to think about it.
你能看出这是怎么办到的吗？如果你想花些时间思考一下，请暂停视频。
Here's the answer. The dark gray square is a2 and the light gray one is b2. The one outlined in blue is c2.
这是答案。深灰色正方形是a2，浅灰色正方形是b2。蓝色画出的正方形是c2。
Each blue outlined square contains the pieces of exactly one dark and one light gray square, proving the Pythagorean theorem again.
每个蓝色画出的正方形正好包含了一个深灰色正方形和一个浅灰色正方形，再次证明了毕达哥拉斯定理。
And if you'd really like to convince yourself,
如果你真的想说服自己，
you could build a turntable with three square boxes of equal depth connected to each other around a right triangle.
你可以建个转台，上面有三个相同深度的正方形盒子，它们考一个直角三角形相连。
If you fill the largest square with water and spin the turntable,
如果你在最大的正方形内装满水，并转动转台，
the water from the large square will perfectly fill the two smaller ones.
最大的正方形内的水会正好装满另外两个小的正方形。
The Pythagorean theorem has more than 350 proofs, and counting, ranging from brilliant to obscure.
毕达哥拉斯定理有超过350个证明，还有更多，从及其聪明的到有些难懂的。
Can you add your own to the mix?
你能提出一个新的证明吗？
