[camjw]

Commented this on another thread:

Okay so take the triangle made by taking the diagonal of the unit square. This has side lengths 1, 1, and c and has area 1/2.

Now, take four of these and arrange them in a square with the side length being c. It would be easier to draw this... basically you stick the right angles in the center. If this isn't clear I can draw a diagram.

Anyway, you just made a square with side length c but since its made of four of those original triangles we know that the area of it is 4 * (1/2) = c^2 so c^2 = 2.

EDIT: made an excalidraw to explain this construction - maybe helpful https://excalidraw.com/#room=2298a8fd232d5f58e8ca,HmUwSqOt6J...

[chengiz]

You're using geoemetrical construction not dissimilar from proving the theorem for a != b. So it's not in the spirit of this new method. No one disputes there are easier methods to prove the theorem.