Geometry provides a natural window into what it is like to do mathematics. In the world of geometry, playful experimentation is often more fruitful than following a procedure, and logic plus a few axioms can open new worlds. Nonetheless, teaching a geometry course in a way that combines both rigor and play can be difficult. Many geometry courses emphasize axiomatics above all. These courses often begin by having students prove trivial results to learn the structure of a proof; interesting problems that require creative insight are postponed until much later in the course. More student-centered approaches encourage students to discover results for themselves, but because students are expected to know a certain amount of material by the end of the course, the "conjecturing" that students do can have a fill-in-the-blanks feel. The Park School of Baltimore mathematics faculty's solution to this dilemma is to teach geometry by "starting in the middle"--that is, by taking facts that students know from eighth grade as a starting point. In this article, the authors describe the types of problems they pose to students at various stages of their study of geometry and the way their thinking develops in response. (Contains 4 figures.)