The first open question came as an extension of our first investigation from Rethinking Proof (de Villiers, 1999). In the dynamic environment provided by GSP, the students summed the measures of the distances to the sides from a point on the interior of an equilateral triangle. By moving the point inside the triangle, students observed that the sum remains constant. The conjecture can be proven, as indicated in the text, by decomposing the equilateral triangle into three triangles, each of which has one of the distances as the length of its altitude (Figure 1).
The title of Figure 1, like others to follow, links to a JavaSketchpad file that allows you to explore the sketch interactively. While these files don’t have the robustness of a full Sketchpad file, they do allow you to get a feel for the investigations.
Note: If you encounter any difficulty with the JavaSketchpad file, see Mays (2003) for explicit instructions.