Note that we are working with an algebraically closed field (or at least we can assume we are without loss of generality).
This is a lemma from my lectures whose proof I don't understand. The claim is that "changing coordinates, we may assume the ratios $(\alpha: \beta)$ are $(1:0), (0:1), (1:-1), (1:-\lambda)$ for some $\lambda \in K \setminus \{0,1\}$." It then goes on to do a simple proof given this information. However, I am stuck at this change of coordinates. I generally understand how such a change of coordinates could be made: consider the lifts to $K^2$ and make a change of coordinates such that two of the vectors are the new basis vectors. Then, as the actual magnitude of the vector does not matter in projective space, we can scale a basis vector such that the 3rd has coordinates $(1:-1)$ (remember that they are distinct ratios), and we get the last is $(1:-\lambda)$ again using the uniqueness of the ratios.
However, I mainly do not understand why a change of coordinates preserves the property of being a square. The proof seems to rely on this unstated fact, and I can't prove this. Isn't a counterexample $u = t^2$ and $v = 2t + 1$, which is square with $(1:1)$ but not $(1:-1)$ or $(0:1)$ for example? Or am I thinking of this totally the wrong way?