One major theme we have had is the geometry of circles and associated constructions of tangent circles (see the constructions in GTC for conics) and orthogonal circles (recent labs and assignments). The concepts of inversion, power of a point, and radical axis play key roles. This was already on the agenda for the recent quiz.

The central figure of this study has been the figure of a pencil of circles and a second pencil of circles orthogonal to the circles of the first pencil. Two circles (or a circle and a line) determine a pencil. An elliptic pencil of circles through points A and B has a hyperbolic pencil as orthogonal pencil. This hyperbolic pencil is the set of Apollonian circles defined by A and B. The orthogonal pencil of a parabolic pencil is another parabolic pencil.

A second theme has been the study of new geometries using models built from our knowledge of circle geometry.

The points are the points of the Euclidean plane with one point at infinity added. Lines and circles are both considered "inversive circles". The "congruence" transformations of this geometry are circle inversions and line reflections and compositions of such transformations. Angles are preserved by these transformations and "inversive circles" are mapped to "inversive circles". (Also, cross-ratio is preserved, but we have not explored that yet.)

This is a non-standard model of Euclidean geometry. The points are the points of the Euclidean plane, with one point O in the plane removes and with the point I at infinity added. DWEG-lines are "inversive circles" through O. Congruence transformations of the geometry are the compositions of inversions in DWEG-lines. Angles are measured by the Euclidean angle measure between circles. Steps of equal distance can be marked off by reflecting in parallel mirrors. Circles with center A through P can be defined by reflections of P in mirrors through A. These circles turn out to be Apollonian circles.

The points are points inside a Euclidean circle C. Lines are arcs or segments of supporting circles or lines which are orthogonal to C. Line reflection is given by inversion in the supporting circles or lines. Angles are measured by the Euclidean angle measure between circles. Steps of equal distance can be marked off by reflecting in parallel mirrors. Circles with center A through P can be defined by reflections of P in mirrors through A. These circles turn out to be Apollonian circles.

We learned about ultra-parallel lines and (critical) parallel lines.

We did not go too deeply into this subject, but we did learn about the concept of angle defect and the additive properties of the defect.

We are in the midst of this study. We have earlier learned the definition of harmonic division and its relation to inversion and also the ruler constructions with a quadrilateral Recently, we have been introduced to central projections and how figures can be transformed by these projections. We have also learned the theorems of Desargues, Pappsu, and Pascal (see Ogilvy).