This is a good illustration to proof #8. There are 3 or 4 derived from Euclid's more general idea.But no, what I meant is exactly the reference I made.

VI.31 tells us that (in your first diagram)

ka² + kb² = kc²

because such and such triangles are similar and two of them add up to the third. You made it much more specific:

aa' + bb' = cc'

(for the same reason, i.e. because two triangles add up to the third) and then showed that due to the similarity

a' = ka, b' = kb, c' = kc

I do not truly know what makes one proof a variant of, or a corroboration on, another, and what makes two proofs independent. Proofs 6, 7, 8 should have been probably one. The reason I have split them is that years ago all I had was just a few proofs. The rest came along over the space of a few years, not all at the same time. At some point it was already hard to make a better system or to change the numbering.