do infinite family of lines in $\mathbb{R}^2$ have a common point by knowing that any three of them have common point?

Question

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!)

Answer 1

Note that any two distinct lines in the family have at most one point in common. Take any three distinct lines $l_1,l_2,l_3$ from the family, and let $v_0$ be their common point. Take any $l$ in the family distinct from these, and consider $\{l_1,l_2,l\}.$

Added: This result is actually true in all $\Bbb R^n$ (vacuously when $n=1,$ since there is no infinite family of lines, in the first place).

Answer 2

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 $x_0$.