Does every plane curve contain a rational point? Does every plane curve contain a rational point?
I think the answer is yes, but I can not prove this. Please help.
However, if it is possible to build a pathological curve - without rational points, then even more interesting question arises - which properties of a curve will imply existence of a rational point? 

The answer is NO. We can show more by only considering straight lines and the fact that rational points are countable.
Choose an arbitrary point $A$ in ${{\mathbb{R}}^{2}}$ whose coordinates are both irrational. The set $L = \left\{ l:A\in l \right\}$ is uncountable, and thus there is no one-to-one mapping from $L$ to ${{\mathbb{Q}}^{2}}$. Hence, there are uncountably many lines going through $A$ which contains no rational points.
If we go further, we can show that ${{\mathbb{R}}^{2}} - {{\mathbb{Q}}^{2}}$ is path-connected.
