Circular billiards can be used as an entry point for students to discover the generalized inscribed angle theorem.
Inspired by Henri Picciotto’s Lab 1.9 on page 18 (PDF: http://www.mathedpage.org/geometry-labs/index.html) on the generalized inscribed angle theorem and N. Chernov and R. Markarian’s “Chaotic Billiards,” Preview available at: http://www.ams.org/bookstore/pspdf/surv-127-prev.pdf.
Fractals are introduced with the hope that the geometry will be compelling and generative. Ultimately, students discover a relationship between scaling, copies and dimension for standard geometric objects and apply this relationship to understand features of fractals. Infinite series are also generated and explored.