Putnam 1987 A1

Curves $A,B,C$ and $D$ are defined in the plane as follows:

$$\begin{aligned}A &= \left\{ (x,y): x^2-y^2 = \frac{x}{x^2+y^2} \right\}, \\ B &= \left\{ (x,y): 2xy + \frac{y}{x^2+y^2} = 3 \right\}, \\ C &= \left\{ (x,y): x^3-3xy^2+3y=1 \right\}, \\ D &= \left\{ (x,y): 3x^2 y - 3x - y^3 = 0\right\}.\end{aligned}$$

Prove that $A \cap B = C \cap D$.