# Putnam Puzzler 2009

**Question:**

What is the maximum number of rational points that can lie
on a circle in the plan whose center is not a rational
point? (a rational point is a point both of whose
coordinates are rational numbers.)

**Answer:**

There are at most two such points. For example, the points
(0,0) and (1,0) lie on a circle with center (1/2,*x*) for
any real number *x*, not necessarily rational.

On the other hand, suppose *P* = (*a*,*b*),
*Q* = (*c*,*d*), and *R* = (*e*,*f*)
are three rational points that lie on a circle. The midpoint
*M* of the side *PQ* is ((*a+c*)/2,(*b+d*)/2),
which is again rational. Moreover, the slope of the line
*PQ* is (*d-b*)/(*c-a*), so the slope of the line
through *M* perpendicular to *PQ* is (*c-a*)/(*b-d*),
which is rational or infinite.

Similarly, if *N* is the midpoint of *QR*, then *N* is a rational
point and the line through *N* perpendicular to *QR*
has rational slope. The center of the circle lies on both of these lines,
so its coordinates (*g*,*h*) satisfy two linear equations with rational
coefficients, say: *Ag + Bh = C* and *Dg+Eh=F*. Moreover,
these equations have a unique solution. That solution then must be:

*g = (CE - BD)/(AE-BD)*

*h = (AF - BC)/(AE-BD)*

by elementary algebra or Cramer's rule. So the center of the circle is rational. This proves the desired result.