Is $x^3-yx^2 = y^3-xy^2$ transitive?

Question

I'm asked to prove that the relation $R$ on $\mathbb{C},$ $xRy \iff x^3-yx^2 = y^3-xy^2$ is an equivalence relation. It's easily shown it's reflexive and symmetric, but I'm having problems with its transitivity. Any tips?

Answer 1

The claim is false. We have $xRy \iff x^3-yx^2 = y^3-xy^2 \iff x^2(x-y)=y^2(y-x)$ $\iff (x^2+y^2)(x-y)=0 \iff x^2+y^2=0\ \mathrm{or}\ x=y$. So for example $1Ri$ and $iR{-1}$ but $1 \not{R} {-1}$.