Each group should create a word document with figures to answer and explain. The goal should be write a few pages that will teach the topic, to be logically and geometrically clear and not just a minimal proof.
1. If A' is the inversion of A in a circle c, why is any circle through A and A' orthogonal to c? Also, show how this is used to construct a circle through a point A orthogonal to two given circles.
2. Define an Apollonian "circle" of A and B and explain why it is in fact a circle (and discuss the terminology issue of why this is not circular, unnecessary, etc). Relate this to the concepts of harmonic division and inversion. Also, show how the angle bisector ratio theorem gives a good construction of the circle through P.
3. Define the concept of power of a point and tell what is the radical axis of two circles, why it is a line, how it is related to orthogonal circles. Also prove the concurrence theorem and discuss and special cases. Show and explain how to construct the circle orthogonal to 3 given circles.
4. Explain what is meant by a pencil of circles. Tell what are the 3 kinds of pencil. Explain why any two circles belong to a unique pencil and why the circles orthogonal to these circles form a pencil also. Show and explain how to start with two given circles and a point P and construct the circle through P that is in the pencil containing the two circles -- do the cases of circles intersecting in 2, or 1, or 0 points.
5. Explain with proofs why the inversion of a line or a circle is a line or a circles. Describe how these results can be unified by adding a point at infinity. Show how to construct the iimage of a figure made of two circles c and d through points A and B and circles e and f orthogonal to c and d, when the figure is inverted in the circle with center A though B.
6. Define Dr. Whatif's geometry and explain how to construct lines, circles, and perpendicular bisectors and why these constructions work and make sense. Include special cases. Explain how that DWEG geometry is really just usual Euclidean geometry transformed by inversion.