This simple proof of the Pythagorean theorem reveals how brilliant Einstein’s mind is. This is my first time reading his proof and immediately amazed.
Starting with a right triangle, draw a perpendicular line from the hypotenuse to the right angle. This divides the triangle into two smaller right triangles.
$$smaller~area + larger~area = original~area$$
Adding the two triangles equals the original triangle.
Lets say:
$$a = hypotenuse~of~smaller~triangle$$
$$b = hypotenuse~of~larger~triangle$$
$$c = hypotenuse~of~original~triangle$$
These triangles are similar in terms of the angles and their sides are in proportion to each other.
Since they’re all similar, each area occupies a fraction, f, of the area of the square of the hypotenuse.
$$smaller~area = fa^2$$ $$larger~area = fb^2$$ $$original~area = fc^2$$
Using all the relationships,
$$fa^2 + fb^2 = fc^2$$
Dividing the above equation by f,
$$a^2 + b^2 = c^2$$
And the Pythagorean theorem has been proved.