Circles in F

In Euclid’s The Elements, a unique circle in R2 is determined by three noncollinear points. This is proven geometrically by constructing a triangle from the three points and showing that the intersection of the perpendicular bisectors of two sides of the triangle gives a point that is equidistant from all three vertices of the triangle [1]. This point is said to define the center of a circle which circumscribes the triangle formed by the points. In our research, we demonstrate that circles can be similarly determined in Fq , the two-dimensional vector space over the finite field Fq . However, the properties of Fq cause some interesting cases to arise. Among these is the possibility for two distinct points to have zero distance. Nevertheless, we were able to show that three distinct noncollinear points which have nonzero distance from each other determine a unique circle of nonzero radius.