Rational point inside a rational polygon

I have the following conjectures.

Conjecture 1:

Hypotheses:

• Let $P = (v_1, v_2, …. v_n)$ be a (convex or concave) polygon drawn on a plane.

• The lengths of the edges $(v_1, v_2)$, $(v_2, v_3)$ … $(v_n, v_1)$ are all rational numbers.

Conclusion:

• There exists a point $x$ inside the polygon with rational coordinates such that the euclidean distances between the pairs $(x,v_1), (x,v_2), … (x,v_n)$ are all rational numbers.

Conjecture 2:

Hypotheses:

• Let $P = (v_1, v_2, …. v_n)$ be a (convex or concave) polygon drawn on a plane.

• The lengths of the edges $(v_1, v_2)$, $(v_2, v_3)$ … $(v_n, v_1)$ are all rational numbers.

• The co-ordinates of the vertices $v_1, v_2, …. v_n$ are all rational numbers.

Conclusion:

• There exists a point $x$ inside the polygon with rational coordinates such that the euclidean distances between the pairs $(x,v_1), (x,v_2), … (x,v_n)$ are all rational numbers.

The above conjectures sound like a very natural topology problems. Note that the Conjecture 1 implies the Conjecture 2.

What I know so far:

1. The above conjectures are true for $n=3$. This follows from the following theorem.

   Theorem: The set of points with rational distances to the vertices of a given triangle with sides of rational length is everywhere dense.

2. Conjecture 1 is false for $n > 3$. For a proof, see Robert Kleinberg’s comment on my blogpost.

Questions about Conjecture 2:

• Is it true for $n=4$ ?

• Is it true for convex polygons ?

• Is it true for convex polygon with $n=4$ ?

• Is it true for any other special cases ?

• Are there any known generalizations to higher dimensions?

A very special case I am very interested in:

• Let $Q = (v_1, v_2, v_3, v_4)$ be a polygon.

• The co-ordinates of the vertices $v_1, v_2, v_3, v_4$ are all rational numbers.

• The lengths of the edges $(v_1, v_2)$, $(v_2, v_3)$, $(v_3, v_4)$ and $(v_4, v_1)$ are all rational numbers.

• The distance between $v_1$ and $v_3$ is rational.

Conjecture Q1: There exists a point $x$ with rational coordinates inside $Q$ such that the euclidean distances between the pairs $(x,v_1), (x,v_2), (x,v_3), (x,v_4)$ are all rational numbers.

Conjecture Q2: Same as Conjecture Q1 when the polygon $Q$ is convex.

Answer

Attribution
Source : Link , Question Author : Shiva Kintali , Answer Author : Community