Reply To: What are brains for??

Not each other! The two ‘ab’s from the first part get canceled by the two ‘-ab’s of the second part. First part being: 2ab, second part being: (b-a)^2.

Well, I guess you need brains. Actually, there is a visual proof to the a^2 + b^2 = c^2 theorum. However, I can’t put it up here. I saw it in a Sefer by Reb Eliyahu of Izmir. The Gilyon Hashas in Menachos references it.

Draw four rectangles at right angles to each other, as above, and form a box. Next, draw a box from all hypotinuses. Also, extend the line from the inner tip of two rectangles until the end, that should give a large box on one corner and a frame on two sides of it. Now, if you take the areas of the squares created by any ‘a’ and ‘b’ from the outer rectangles, you’ll see that its pieces match those of the slanted square created by the hypotenuses.

Is a thousand words worth a picture?