We solve the following problem. For a given rational circle C passing through the rational points p, q, r and a given angle α, compute a rational point on C whose angle at C differs from α by a value of at most ϵ. In addition, try to minimize the bit length of the computed point. This document contains the C++ program |rational_points_on_circle.c|. We use the literate programming tool |noweb| by Norman Ramsey