Consider a right-angled triangle with vertex co-ordinates $(0,0), (a,0)$ and $(0,b)$. The length of the hypotenuse is $c$. Moreover, $a, b$ and $c$ are positive integers i.e., $(a,b,c)$ is a Pythagorean triple.
Following are my conjectures in the increasing order of difficulty.
Conjecture 1: There exists a rational point $(r,0)$ at a rational distance from $(0,b)$, such that $r > 0$, $r \neq a$.
Conjecture 2: There exists a rational point $(r,0)$ at a rational distance from $(0,b)$, such that $0< r < a$.
Conjecture 3: There exists an infinite number of rational points $(r, 0)$ at a rational distance from $(0,b)$, such that $0< r < a$.
Conjecture 4a: For any two rational numbers $0< r_1 < r_2 < a$, there exists a rational point $(r,0)$ at a rational distance from $(0,b)$, such that $0< r_1 < r < r_2 < a$.
Conjecture 4b: For any two rational numbers $0< r_1 < r_2 < a$, there exists an infinite number of rational points $(r,0)$ at a rational distance from $(0,b)$, such that $0< r_1 < r < r_2 < a$.
Note that conjecture 4a implies 4b.
My question: Are the above conjectures known to be true / false ?
Note: In all the above conjectures, I want the existence of the point $(r,0)$ on the $x$-axis, for both cases $a > b$ and $a < b$.
Motivation: An affirmative answer to the above questions will help me generate certain fractals in a clean way.
A known fact: The set of points with rational distances to the vertices of a given triangle with sides of rational length is everywhere dense 1.
1 J.H.J. Almering, Rational quadrilaterals, Indag. Math. 25 (1963) 192–199.