2023-07-10

The 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.

there are many proofs for the famous Pythagorean theorem. A Babylonian tablet from around 1800 B.C. lists 15 sets of numbers that satisfy this theory.The theory is that surveyors could stretch a knotted rope with twelve equal nots to form a triangle with sides of length 3, 4 and 5.Some historians think that Ancient Egyptian surveyors used one such set of numbers,such as 3, 4, 5, to make corners.

The earliest known Indian 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 came from the Pythagorean theorem.

But how do we know that the theorem is true for every right triangle on a flat surface? Because we can prove it.

One classic proof is 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 are a² and b².

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 Greek mathematician Euclid and was also created almost 2, 000 years later by twelve-year-old Einstein. This proof says 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. rearrange the terms，and finally, add the two equations together and simplify to get ab²+ac²=bc², or a²+b²=c².

The dark gray square is a² and the light gray one is b². The one outlined in blue is c². Each blue outlined square contains the pieces of one dark and one light gray square, proving the Pythagorean theorem again.

You could even 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 they are all very fun to learn. Can you add your own prof?

This statement is one of the most important rules of geometry, and the basis for constructing stable buildings and GPS.