Recently my colleague and I were thinking of a coordinate geometry proof on Pythagoras' Theorem with Euclid's idea but two squares only. But we are not sure whether it is an old idea or not and would like to share to you guys.
The proof is as follows:
It is known that O is the origin, OADH and OFED are two squares. The coordinates of D are (a,b). By rotating D anti-clockwise 90 degrees about O to F, we have F (-b,a) and rotating E clockwise 90 degrees about F to O, we have E (a-b, a+b).
Now, the area of the square OADH is b^2. Note that triangle OHF and triangle OAD are congruent (or image by rotation), and area of triangle OHF is half of the area of the square OADH, while area of triangle OAD is half of the area of the rectangle OBCD, so the area of the rectangle OBCD = area of the square OADH = b^2.
Similarly, we can verify the area of the rectangle BFEC is a^2. So, combining we have OF^2 = a^2+b^2 = OA^2 + AF^2. QED
Please let us know if you have any further ideas or comments on this proof!