singingbanana's video: A Pythagorean Theorem for Pentagons Einstein s Proof

Pythagoras's Theorem is the most famous theorem in mathematics, commonly stated as "the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides." However, Pythagoras's Theorem is not just for squares. In fact it works for any shape. The proof relies on the fact that scaling a shape by c will scale the area by c^2. Then, if Pythagoras's Theorem is true then the area of the shape on the hypotenuse will be equal to the total areas of the similar shapes on the other two sides. More succinctly, if Pythagoras's Theorem is true then the areas will be equal. But we can prove Pythagoras's Theorem itself by running that argument in reverse - if the shapes have equal area then Pythagoras's Theorem is true. This is an argument an 11 year old Albert Einstein used to prove Pythagoras's Theorem for himself. There are a couple of things I wished I said clearer in the Einstein proof: The Einstein proof divides the triangle so we have three right-angled triangles (but I think that was clear from the picture) Secondly, the three triangles are scaled versions of a triangle with a hypotenuse of length 1 and area X, which then have areas scaled by a^2, b^2 and c^2. (I just said "some triangle"). This topic has been done before by a couple of the big maths YouTube channels, which I didn't know at the time (or forgot). Numberphile did it in 2014 https://www.youtube.com/watch?v=ItiFO5y36kw And Mathologer did it in 2018 https://www.youtube.com/watch?v=p-0SOWbzUYI A little historical note, Pythagoras's Theorem appears twice in Euclid's Elements, the famous squares version appears in Book 1.47, and in Book 6.31 it is there again, this time for any shape.