In Example 1.23, the author argues that an hypersurface $X \subset \mathbb{A}^{3}$ generated by $x^{3}+y^{3}+z^{3}=1$ is rational. To see this, let $L_{0}: x+y=0,z=1$ and $L_{1}: x+\epsilon y=0, z=\epsilon$ be two lines where $\epsilon$ is nontrivial 3rd roots of unity. Then, $L_{0}$ and $L_{1}$ are inside of $X$. Now the author claims that for any $x \in X$, there exists the unique line pasing through $x$ and intersecting with both $L_{0}$ and $L_{1}$.
My question is, how can we know that such a line is unique? Any hint or approach will be appreciated.
By hint of @KReiser, I proved as below;
Let $x=(a,b,c)$ in $X \setminus (L_{0},L_{1})$. Then, using cross product, the plane containing $L_{0}$ and $x$ is $(1-c)x +(1-c)y+(a+b)(z-1)=0$. If we parametrize $L_{1}$ by $(-\epsilon t, t, \epsilon)$, then intersection occur when $t= \frac{a+b}{1-c}$ with $c \neq 1$. Since $c=1$ means $x \in L_{0}$, contradiction, $t$ is well-defined. Thus, we have a line containing $x$ and meeting $L_{1}$ and $L_{2}$ by drawing a line between $L_{1}(\frac{a+b}{1-c})$ and $x$. Conversely, if such a line $L$ containing $x$ and meeting $L_{0}$ and $L_{1}$ occur, then the line $L$ and $L_{0}$ form a plane containing $x$ and $L_{0}$ thus by above construction, this line $L$ should contain $L_{1}(\frac{a+b}{1-c})$. This shows that the existence of such line for $x$ is unique.