I am working on a geometry problem and it lead me to make the following conjecture:
Conjecture 1: Given any convex polygon (v1, v2, …. v_n) drawn on a plane with integer edge lengths, there is a point x inside the polygon such that the euclidean distances between the pairs (x, v1), (x, v2), … (x, v_n) are all rational numbers.
Theorem: Conjecture 1 is true for n = 3.
Simple Homework: Prove Conjecture 1 for an equilateral triangle.
Conjecture 1 seems like a very natural geometry problem. Is it well-known ? If you know any references, please leave a comment or email me.