The earliest known proof of the Pythagors theorem was provided by the ancient Indian mathematicians, thousands of years before Pythagoras. Pythagoras himself is said to have visited India and learnt the theorem there, but returned to Greece without learning the proof of this theorem and hence has presented only the theorem and not its proof.

Here is an Ancient Chinese proof of this so called Pythagoras Theorem.
Consider the following diagram:

If you see the Square ABCD as slightly rotated in the above diagram then it is just an illusion. Any way, the proof given by the Chinese based on the above diagram is as follows.

ABCD is a square with the length of each side being (a+b). Hence,

Area of Square ABCD = (a+b)² ( eq. 1 )

Now consider another square inscribed inside ABCD i.e. EFGH whose length of the sides is c. (Look at the above diagram). Also there are 4 right angled triangles (every angle in a square is a right angle) inside the square ABCD. We also see that the area of all these 4 right angled triangles is equal and is given by 1/2 x b x a.

Area of each right angled triangle = (1/2) x a x b                 ( eq. 2 )

We can also see that,

Area of the Square EFGH = c² ( eq. 3)

Now we see that the Area of the Square ABCD is equal to the Sum of the Areas of the 4 right angled triangles and the Area of the square EFGH. i.e.

Area of Square ABCD = (4 x Area of a right angled trianlge) + Area of Square EFGH

Substituting equations 1,2 and 3 in the above equation we get

=>          (a+b)² = [4 x (1/2) x a x b] + [ c² ]

=>          (a+b)² = 2ab + c²

=>          a² + b² + 2ab = 2ab + c²

=>          a² + b² = c²

This is nothing but the theorem what we know today in the name of Pythagoras! This proof clearly shows that in a right angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.

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