Our interest in this chapter is in circular inversion and the nine point circle, constructions in planar Euclidean geometry that were developed after the Arabic era. We assume all of Hilbert’s axioms and their consequences as expressed in Book I of The Elements. Circular inversion, a lovely topic in its own right, proves useful in understanding models of the hyperbolic plane. The nine point circle, which is about triangles and the lines, centers, and circles naturally associated to triangles, typifies work done in Euclidean geometry in Europe during the eighteenth and nineteenth centuries. With circular inversion and the nine point circle as goals, our journey winds naturally through some favorite classical ideas in Euclidean geometry.