Two questions are asked that concern the work of teaching high school geometry with problems and engaging students in building a reasoned conjecture: What kinds of negotiation are needed in order to engage students in such activity? How do those negotiations impact the mathematical activity in which students participate? A teacher's work is analyzed in two classes with an area problem designed to bring about and prove a conjecture about the relationship between the medians and area of a triangle. The article stresses that to understand the conditions of possibility to teach geometry with problems, questions of epistemological and instructional nature need to be asked--not only whether and how certain ideas can be conceived by students as they work on a problem but also whether and how the kind of activity that will allow such conception can be summoned by customary ways of transacting work for knowledge. (Contains 8 figures and 12 footnotes.)