Pythagorean Theorem by Homothetic Copies
This one was sent to me by Geoffrey Margrave from Lucent Technologies. It looks very much as #8, but is arrived at in a different way. Create 3 scaled copies of the triangle with sides a, b, c by multiplying it by a, b, and c in turn. Put together, the three similar triangles thus obtained to form a rectangle whose upper side is
In this form the proof appears in [Birkhoff, p. 92].
Yet another variant that could be related to #8 has been sent by James F.:
The latter has a twin with a and b swapping their roles.