Michael Williams and Dr. Ivona Grzegorczyk
The inscribed angle theorem tells us that an angle inscribed in a circle has one half the measure of the central angle of the circle that subtends the same arc. Another way to think of this is that all the points which view a segment at a constant angle will lie on the arc of a circle or its reflection. This statement has been generalized to account for two segments and results in a curve known as the Apollonian cubic. This research further generalizes the Apollonian cubic to a curve with arbitrary angle. The main question is, what are the points that view two segments with oriented angles having constant sum? Using techniques from complex analysis and ideas from inversive geometry we find, in general, that the points lie on the circle inversion of a Cassini Oval. Cassini Ovals are important curves in mathematics with applications in astronomy, nuclear physics, acoustics, and other sciences.