A response to MathAdam’s proof of the Pythagorean Theorem

I was going to post this in the comments below Adam’s article on the paper folding proof of the Pythagorean Theorem, but I didn’t see how to post images. So here it is. I thought up something equivalent to this proof during an 8 mile straight stretch in my drive to school about a decade ago. By the time I reached the stop light at the end of the stretch I had it nailed down in my head. In the intervening years I forgot the details, but in reconstructing it for this response a simpler version came to me.

It relies on the SOH-CAH-TOA definitions of the trig functions, but only uses them as a way of organizing statements about the ratios of the sides of the triangles.

Note:

— The three triangles are similar.

— a sin(theta) + b cos(theta) =c

— sin(theta)=a/c; cos(theta)=b/c

— Substitute: a(a/c) + b(b/c) =c

— Therefore a² + b² =c².