# Rational point inside a rational polygon

I have the following conjectures.

Conjecture 1:

Hypotheses:

• Let $$P=(v1,v2,….vn)P = (v_1, v_2, …. v_n)$$ be a (convex or concave) polygon drawn on a plane.

• The lengths of the edges $$(v1,v2)(v_1, v_2)$$, $$(v2,v3)(v_2, v_3)$$$$(vn,v1)(v_n, v_1)$$ are all rational numbers.

Conclusion:

• There exists a point $$xx$$ inside the polygon with rational coordinates such that the euclidean distances between the pairs $$(x,v1),(x,v2),…(x,vn)(x,v_1), (x,v_2), … (x,v_n)$$ are all rational numbers.

Conjecture 2:

Hypotheses:

• Let $$P=(v1,v2,….vn)P = (v_1, v_2, …. v_n)$$ be a (convex or concave) polygon drawn on a plane.

• The lengths of the edges $$(v1,v2)(v_1, v_2)$$, $$(v2,v3)(v_2, v_3)$$$$(vn,v1)(v_n, v_1)$$ are all rational numbers.

• The co-ordinates of the vertices $$v1,v2,….vnv_1, v_2, …. v_n$$ are all rational numbers.

Conclusion:

• There exists a point $$xx$$ inside the polygon with rational coordinates such that the euclidean distances between the pairs $$(x,v1),(x,v2),…(x,vn)(x,v_1), (x,v_2), … (x,v_n)$$ 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 $$n=3n=3$$. 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 $$n>3n > 3$$. For a proof, see Robert Kleinberg’s comment on my blogpost.

• Is it true for $$n=4n=4$$ ?

• Is it true for convex polygons ?

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

• 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 $$Q=(v1,v2,v3,v4)Q = (v_1, v_2, v_3, v_4)$$ be a polygon.

• The co-ordinates of the vertices $$v1,v2,v3,v4v_1, v_2, v_3, v_4$$ are all rational numbers.

• The lengths of the edges $$(v1,v2)(v_1, v_2)$$, $$(v2,v3)(v_2, v_3)$$, $$(v3,v4)(v_3, v_4)$$ and $$(v4,v1)(v_4, v_1)$$ are all rational numbers.

• The distance between $$v1v_1$$ and $$v3v_3$$ is rational.

Conjecture Q1: There exists a point $$xx$$ with rational coordinates inside $$QQ$$ such that the euclidean distances between the pairs $$(x,v1),(x,v2),(x,v3),(x,v4)(x,v_1), (x,v_2), (x,v_3), (x,v_4)$$ are all rational numbers.

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