While teaching a methods class for preservice secondary school mathematics teachers, Christopher E. Smith found that although all students could draw a reasonably close approximation of a circle, not all could provide an entirely accurate definition of a circle. A discussion with students led him to think about ways of reintroducing students to the idea of a locus and the notion of a circle as a set of points rather than--or perhaps in addition to--the idea of a circle as a single object or image. This realization inspired him to introduce students to the non-Euclidean geometry system known as taxicab geometry and to explore some familiar shapes, although in forms that would make them question how they think about these objects as well as their definitions. The lesson described in this article was used with college students, but the motivation for teaching it and the ideas presented are applicable to classrooms with younger students as well, either as an introduction to forthcoming concepts or as a review before transitioning from a more synthetic approach to a more analytic approach.