In fall 2013 the author taught a graduate course entitled Topics in Geometry for Middle and High School Teachers. The majority of the participants had extensive experience teaching mathematics in middle school or high school. The fact that participants were rather comfortable with the basic concepts of school geometry made it both possible and necessary to work on questions that would require making deeper connections among familiar ideas. The discussion presented in this article evolved aroundone of the questions assigned for homework: Compare the sum of the medians of a triangle with the triangle's perimeter. The author hoped that all the participants would draw clear, accurate diagrams and would formulate observations that would create a solid foundation for a productive classroom discussion.