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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.

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    $\begingroup$ Would the theorem be more impressive if you scaled distances and placed all points on a a grid in $\mathbb{Z}^2$? $\endgroup$ Oct 5, 2021 at 15:50
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    $\begingroup$ I can't see that topology is involved here. $\endgroup$
    – Paul Frost
    Oct 6, 2021 at 10:07
  • $\begingroup$ Can you clarify which are the hypotheses and which are the conclusions of your conjecture? $\endgroup$ Oct 6, 2021 at 16:39
  • $\begingroup$ @DavidG.Stork There are examples of polygons with integer edge lengths with no integer point x. But a rational point x exists. $\endgroup$ Oct 6, 2021 at 17:56
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    $\begingroup$ D19 in Richard Guy, Unsolved Problems in Number Theory: "Is there a point all of whose distances from the corners of the unit square are rational?" I believe this problem remains unsolved. Guy gives much discussion of this & related questions, and also many references to the literature. $\endgroup$ Oct 6, 2021 at 23:49

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