Rational point inside a rational polygon

I have the following conjectures.

Conjecture 1:

Hypotheses:

  • Let P=(v1,v2,.vn) be a (convex or concave) polygon drawn on a plane.

  • The lengths of the edges (v1,v2), (v2,v3)(vn,v1) 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,v1),(x,v2),(x,vn) are all rational numbers.

Conjecture 2:

Hypotheses:

  • Let P=(v1,v2,.vn) be a (convex or concave) polygon drawn on a plane.

  • The lengths of the edges (v1,v2), (v2,v3)(vn,v1) are all rational numbers.

  • The co-ordinates of the vertices v1,v2,.vn 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,v1),(x,v2),(x,vn) 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=(v1,v2,v3,v4) be a polygon.

  • The co-ordinates of the vertices v1,v2,v3,v4 are all rational numbers.

  • The lengths of the edges (v1,v2), (v2,v3), (v3,v4) and (v4,v1) are all rational numbers.

  • The distance between v1 and v3 is rational.

Conjecture Q1: There exists a point x with rational coordinates inside Q such that the euclidean distances between the pairs (x,v1),(x,v2),(x,v3),(x,v4) 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

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