Let $x=x(y,z)$, $y=y(x,z)$, $z=z(x,y)$ be implicit defined by the equation $F(x,y,z)=0$. Suppose $x_0=x(y_0,z_0), y_0=y(x_0,z_0), z_0=(x_0,y_0)$ and that at the point $(x_0,y_0,z_0)$ the partial derivatives of $F$ are different from zero.
Show that: $$\dfrac{\partial x}{\partial y}{\big|}_{{y=y_0}\\{z=z_0}}\cdot \dfrac{\partial y}{\partial z}{\big|}_{{x=x_0}\\{z=z_0}}\cdot \dfrac{\partial z}{\partial x}{\big|}_{{x=x_0}\\{y=y_0}}= -1$$
I tried just to calculate the partial derivatives and see what happens, but I couldn't get anywhere.
The first thing you should try to establish is that $$ \left.\frac{\partial z}{\partial x}\right|_{x=x_0,y=y_0,z=z_0} = - \frac{\partial F/\partial x(x_0,y_0,z_0)}{\partial F/\partial z(x_0,y_0,z_0)} $$ From there, your equation can by proved by cycling the $x$, $y$, and $z$ variables.