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[tex]xRy[/tex] iff [tex]x\bar{y}=\bar{x}y[/tex]

It is an equivalence relation and so by letting x=a+bi and y=c+di, then it is equivalent to a/b=c/d so I was viewing it as partitioning points in [tex]\mathbb{C}[/tex] by drawing lines through the origin and equivalent points lie on the line, but rearranging a/b=c/d gives the 2x2 determinant formula (zero case) so I was wondering whether I'm missing something here, is there some other way to think about this, some other possible insight? It seems rather curious that it comes out like the determinant...