Finding rational points at rational distance in the plane

Question

Take any point $p$ in the real plane.

Does there always exist a rational point at a rational distance from $p$?

(A rational point is a point $(q,r)$ where $q$ and $r$ are rational.)

Answer 1

Let our point $P$ be $(\pi, 0)$. If the rational point $(a,b)$ were at a rational distance from $(\pi,0)$, then $\pi$ would be algebraic. But it is not.

Detail: If the distance from $(x,0)$ to $(a,b)$ is $r$, then $x^2-2ax +a^2+b^2-r^2=0$. If $a$, $b$, and $r$ are rational, then $x$ is a root of a quadratic equation with rational coefficients. But $\pi$ is not the root of a non-trivial polynomial equation with rational coefficients.

A somewhat fancier argument shows that there is also no rational point at a rational distance from $(2^{1/3},0)$.