Let $P=(v_1, v_2, v_3, v_4)$ be a simple polygon drawn on a plane.

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

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

  • The distance between $v_1$ and $v_3$ is an integer.

Conjecture 1: There exists a point $x$ with rational coordinates inside $P$ such that the euclidean distances between the pairs $(x, v_1), (x, v_2), (x, v_3)$ and $(x, v_4)$ are all rational numbers.

Conjecture 2: There exists infinite such points $x$ in $P$.

Conjecture 3: All such points $x$ in $P$ are dense in the plane of the polygon $P$.

Conjecture 4: Above conjectures when the polygon $P$ is convex.

My Questions:

  • Are the above conjectures true ?

  • Are there any similar known conjectures or results ?



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