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.

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The conjecture is false for n>3. I don’t know whether there is a counterexample for n=3.

For n>3, the conjecture can be disproven by a counting argument. There are uncountably many (non-congruent) convex n-gons with given side lengths, and only countably many of them contain a point x whose distances from v1, v2, v3 are all rational. To see this, consider v1 and v2 as fixed, while the positions of the other vertices v3, v4, …, vn are variable. A linkage made of n>3 rods of specified lengths is not rigid, so if there is at least one convex n-gon with the specified side lengths, then there must be uncountably many points v3 such that v1, v2, v3 are the first three vertices of a convex n-gon with the specified side lengths. However there are only countably many ordered triples of rational numbers (p,q,r). For any one of these triples, there are at most 2 points x satisfying d(x,v1)=p and d(x,v2)=q, and for each such x there are at most 2 points v3 satisfying d(x,v3)=r and d(v2,v3)=s, where s is the specified length of side (v2,v3).

Thanks Bob. Luckily, I need the Conjecture 1 to be true only when the co-ordinates of the vertices of the convex polygon are rational numbers.

Conjecture 2: Same as conjecture 1, when the co-ordinates of the vertices of the convex polygon are rational numbers.

Conjecture 2 sounds like a very natural topology question.

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