Rational Points On a Circle

Elsewhere we found that there are infinitely many rational points on a unit circle centered at the origin of a Cartesian coordinate system. By a rational translation and/or a dilation, any circle with rational radius and center carries infinitely many rational points. We further saw that circles with irrational centers may not have rational points at all. At the 69th Putnam Mathematical Competition, problem B1 posed a related question:

What is the maximum number of rational points that can lie on a circle in R² whose center is not a rational point? (A rational point is a point both of whose coordinates are rational numbers.)

Solution

Solution

The answer to the question is 2: there are at most two rational points on a circle with irrational center. Before proving the result, let's recollect a few fundamental facts.

For any pair of rational points, the straight line passing through the two may be expressed by an equation with rational (and, hence, integer) coefficients. The midpoint of the segment joining two rational points is itself a rational point. A line perpendicular to a line with slope α has a slope -1/α, implying that the slopes of the two lines are either both rational or both irrational. Therefore, the perpendicular bisector of a segment joining two rational points always has an equation with rational coefficients. The solution to a system of two linear equations with rational coefficients, if exists, is necessarily rational. Which is to say, that if two straight lines that are expressed by linear equations with rational coefficients intersect, their point of intersection is necessarily rational.

With these preliminaries, we are ready to confront the problem. Assume to the contrary there is a circle with irrational center on which there are three distinct rational points, say, P, Q, R. The perpendicular bisectors of PQ and QR would meet at the center of the circle which, from the foregoing discussion, would be a rational point in contradiction with the conditions of the problem.

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