This exercise comes from Shafarevich II.3.6.
Let $\varphi:\mathbb{P}^2 \to \mathbb{P}^2$ be the rational map defined by $\varphi(x_0 : x_1 : x_2) = (x_1x_2:x_0x_2:x_0x_1)$. Consider the point $x = (1:0:0)$ and a curve $C$ that is nonsingular at $x$. The map $\varphi$ restricted to $C$ is regular at $x$ (since the codimension of non-regular points is at least 2), and maps $x$ to some point $\varphi_C(x)$. Prove that $\varphi_{C_1}(x) = \varphi_{C_2}(x)$ if and only if the two curves $C_1, C_2$ touch at $x$ (that is, their tangent spaces are equal at $x$).
I'm having trouble getting a handle on this problem. I want to say something like, "this map maps $x$ to the tangent line at $x$," but I don't know how to view it as such if I don't know anything about the curve.
Another thing I've though about, but haven't been able to make headway on, is blowing up $\mathbb{P}^2$ at $x$, and trying to say something about the preimage of the tangent line to $C_1$.