Was Pythagoras Theorem actually proved by Pythagoras?

With all respect to Pythagoras, the so called Pythagoras’ Theorem was known to the ancient Indians long before the time of Pythagoras. Although Pythagoras introduced his theorem to the Western mathematical and scientific world long after, yet that theorem continues to be known as Pythagoras’ Theorem!

There are several proofs given by Indian mathematicians, everyone of which is much simpler than given by Pythagoras or Euclid. Here, we’re going to discuss few such proofs of the Pythagoras theorem:

First Proof of Pythagoras Theorem

a2 + b2 = c2

Where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides of a right angled triangle.

Here, AE pythagoras theorem= BF = CG = DH = x and ED = HC = BG = FA = y

Now, the square ABCD = square EFGH + the 4 congruent right angled triangles around it.

Therefore, z2 + 4 (1/2*x*y) = (x+y)2 or, z2 = x2 + y2. Hence proved.

Second Proof of Pythagoras Theorem

Here we’ll use the property of similar triangles that the area of similar triangles is proportional to the squares of homologous sides.

Any right triangle can be split into two similar smaller right triangles by drawing a perpendicular from point of the right angle. The similarity can be checked using Angle-Angle-Angle similarity.

It’s also evident from the diagram that: similar triangles

Area (Big) =  Area (Medium) + Area (Small)

Now since the triangles are similar they will have the same area equation.

If the long side is c (5), the middle side b (4), and the small side a (3), our area equation for these triangles is:


Area = F * hypotenuse^2   where F is some area factor (6/25 or .24 in this case; the exact number doesn’t matter).Now let’s play with the equation:

Area (Big) = Area (Medium) + Area (Small)

F c^2 = F b^2 + F a^2

Divide by F on both sides and you get:

c^2 = b^2 + a^2

Hence we get the famous Pythagoras theorem.

There are more than 100 ways of proving Pythagoras theorem. Which one is your favorite proof?

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