Thursday, June 25, 2020

Pythagorean theorem redux

Here is a cool proof I saw on Twitter, it was an RT by @StevenStrogatz from this guy:

Take a generic right triangle.  Flip, rotate and scale it by multiplying each side by a factor of b.  I did this by imagining that a = 1 and then b = ab.  The complementary angles are marked with circles on the right.

So then construct a rotated triangle from the same input, but scaled by a factor of a, and attach it to the other one.  We know that the sides marked ab are parallel, and that the angle between bc and ac is a right angle, by the properties of right, complementary and supplementary angles.

The four outside vertices form a rectangle, from the angles and also since ab = ab.

Finally, rotate and scale again, by a factor of c:

Slide them together