Can a Rational Point Always be Found Amongst a Set of Points in the XY-Plane?

[Dragonfall]

Given n points n1,...,nk in the xy-plane, is it always possible to find a point p such that d(ni,p) is rational for 0<i<k+1?

 

[HallsofIvy]

What if one point is (x₁,0), with x₁ rational, and the other is (x₂,0), with x₂ irrational?

 

[Dragonfall]

Draw a line segment AB between (x1,0) and (x2,0) and a line L bisecting the line segment AB perpenticularly.

 

[daveb]

um...either I'm misinterpreting the OP or the answer can be seen by drawing a circle radius p/q (where p/q is rational) around any of the points (with the point as the center).

 

[HallsofIvy]

um...either I'm misinterpreting the OP or the answer can be seen by drawing a circle radius p/q (where p/q is rational) around any of the points (with the point as the center).

Perhaps you are misinterpreting. The question was whether, given a finite set of points, there exist a point p such that its distance to every point in the set is rational. Certainly every point on the circumference of your circle has rational distance (p/q) from the center, but what about the other points in the set?

 

[Dragonfall]

I've tried everything I know. I don't know how to produce an answer.

 

[StatusX]

Was this given to you as an assignment, or did you just think of it yourself? It may be a much deeper question than it appears.

 

[Dragonfall]

A friend sent this 'funny problem' that he got from a 'funny book'. I brought it to the Canadian undergraduate math conference last week and everyone was stumped.

EDIT: Oh and d is the Euclidean metric. No cheating.