Creating Mathematical Experience in the Classroom - A Surprise Ending

With most of the class satisfied that we now understood the underlying reasons that the reflected points returned, the last day of class provided a dramatic lesson in the complexity of mathematical relationships. P. Henson came in with a result that a general point on the side of an arbitrary pentagon has a return time of 10. The proof followed the method of C. Rojas. What was so shocking about this result is that the angle bisectors are distinctly not concurrent, and the reflections are not co-circular. (See Figure 6.)

The class ended with a sense that we had uncovered more questions than we answered. This was a fitting ending to the class project, as this final realization is a common feature of mathematics research.

Figure 6: Reflection over Angle Bisectors in an Irregular Pentagon  

Click on the title for the JavaSketchpad version. If you have Geometer's Sketchpad, you can download and run the GSP file.