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.

• 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.

• Would the theorem be more impressive if you scaled distances and placed all points on a a grid in $\mathbb{Z}^2$? Oct 5, 2021 at 15:50
• I can't see that topology is involved here. Oct 6, 2021 at 10:07
• Can you clarify which are the hypotheses and which are the conclusions of your conjecture? Oct 6, 2021 at 16:39
• @DavidG.Stork There are examples of polygons with integer edge lengths with no integer point x. But a rational point x exists. Oct 6, 2021 at 17:56
• 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. Oct 6, 2021 at 23:49