This question is a generalization of my previous question.
Consider the two right-angled triangles shown in the above picture. The first triangle has vertex co-ordinates $(0,0), (a,0)$ and $(0,b_1)$. The second triangle has vertex co-ordinates $(0,0), (a,0)$ and $(0,b_2)$. The length of the hypotenuses of these two triangles are $c_1$ and $c_2$ respectively. Moreover, $a, b_1, b_2, c_1$ and $c_2$ are positive integers i.e., $(a,b_1,c)$ and $(a,b_2,c)$ are Pythagorean triples.
Following are my conjectures in the increasing order of difficulty.
Conjecture 1: There exists a rational point $(r,0)$ at a rational distance from $(0,b_1)$ and $(0,b_2)$, such that $r > 0$, $r \neq a$.
Conjecture 2: There exists a rational point $(r,0)$ at a rational distance from $(0,b_1)$ and $(0,b_2)$, such that $0< r < a$.
Conjecture 3: There exists an infinite number of rational points $(r, 0)$ at a rational distance from $(0,b_1)$ and $(0,b_2)$, such that $0< r < a$.
Conjecture 4: For any two rational numbers $0< r_1 < r_2 < a$, there exists a rational point $(r,0)$ at a rational distance from $(0,b_1)$ and $(0,b_2)$, such that $0< r_1 < r < r_2 < a$.
My question: Are the above conjectures known to be true / false ?
Note: In all the above conjectures, I would like to know the existence / non-existence of the point $(r,0)$ on the $x$-axis, for all the three cases: (i) $a < b_1$, (ii) $b_1 < a < b_2$ and (iii) $a > b_2$.
Motivation: An affirmative answer to the above questions will help me generate certain fractals in a clean way.