It is proved that in any Euclidean plane in which  and L are given (possibly equal) lines and P is a given point, then the number of circles in  such that is tangent to  the center of  is on L, and  passes through P is either 0, 1, 2, or  Examples are given to show that each of these possibilities can be realized. This note could be used as enrichment material in a precalculus course.

Euclidean plane, circle, line, point, center, tangent line, rotation, translation, quadratic formula, discriminant.