Derivation of Pythagoras Theorem

The Pythagoras Theorem is an extremely important property in geometry that used been useful in Physics and in various engineering disciplines. Originally discovered by Pythagoras in ancient Greece around 550 B.C, Pythagoras Theorem is independently uncovered in other civilisations, such as China and India. One such intuitive derivation, in this link, is credited to the ancient Chinese mathematicians. However, the complete technical proof of the theorem is only first discovered by Euclid, another Greek philosopher, 250 years after Pythagoras. I will attempt to illustrate the proof below:

*Refer to the below diagram.

Pythagoras Theorem: To show that, for any right-angle triangle, a^2 + b^2 = c^2

From a Triangle ABC, let us extend a square out of each side of the triangle, such that:

Area of Square MNBA = a^2

Area of Square BEDC = b^2

Area of Square ACYX = c^2

Next, let us create lines BT, MC and BX. And observe the 2 triangles MAC and BAX.

Angle MAC = Angle BAX (since Angle BAC is a shared angle)

Length AC (of Triangle MAC) = Length AX (of Triangle BAX)

Length AM (of Triangle MAC) = Length AB (of Triangle BAX)

Hence, Triangle MAC and Triangle BAX are congruent by Side-Angle-Side (SAS).

Next, note that:

Area of Triangle MAC = 1/2 x Area of Square MNBA

Area of Triangle BAX = 1/2 x Area of Rectangle ASTX

Hence, Area of Rectangle ASTX = Area of Square MNBA = a^2 (Red Region)

Similarly, we use the same process to prove that:

Area of Rectangle SCYT = Area of Square BEDC = b^c (Blue Region)

Therefore, we have proven a^2 + b^2 = c^2.