Pythagorean Theorem Proof
Theorem
Given a right-angled triangle, if we call the length of the hypotenuse $c$, and the lengths of the other two sides $a,b$, then the following equation holds. $$ a^2 + b^2 = c^2 $$
Explanation
Apart from its wide applications, this theorem is very practical in itself. It’s named after Pythagoras for leaving behind the oldest ‘proof’, but it is speculated that most ancient civilizations, which could be considered to have formed true societies, were aware of the fact itself.
There are known to be over 400 different proofs of Pythagoras’ theorem. Among them, let’s learn about the oldest theorem, that is, the proof left by Pythagoras himself.
Proof
The length of one side of the outer square is $(a+b)$, and that of the inner square is $c$. The area of the outer square is $(a+b)^2 = a^2 + 2ab + b^2$. The area of the right-angled triangle at each vertex is $\displaystyle {ab \over 2}$. Therefore, we can say that the area of the inner square is $$ (a+b)^2 - 4{ab \over 2} = a^2 + 2ab + b^2 - 2ab $$. Meanwhile, the area of the inner square is also $c^2$, hence $$ a^2 + b^2 = c^2 $$.
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Some have summarized this proof as “just look”. It’s that intuitive and easy, so make sure to see it properly and not forget.