do infinite family of lines in $\mathbb{R}^2$ have a common point by knowing that any three of them have common point? Suppose we have given an infinite family of lines; say $\mathfrak{F}$, in the plane $\mathbb{R}^2$ such that any three of the lines in $\mathfrak{F}$ have a common point. How can we prove that all lines in $\mathfrak{F}$ have a common point. 
(here we should note that using Helly theorem is not applicable, because first of all we are working on an infinite set of convex sets and second that no of them are closed and bounded!)

Pick $L_1,L_2,L_3$. These have a common point $x_0$. In particular,
$L_1, L_2$ have $x_0$ as a common point.
Now pick any other line $L$ in the collection. Since $L_1,L_2,L$ have a common point, it must be $x_0$.
Hence all the lines pass through $x_0$.
