In this thesis, we will formulate and prove Casey’s theorem on the tangency of four circles to a fifth, as well as some of its consequences. For easier understanding, we will start with Ptolemy’s theorem, which deals with the chord quadrilateral and is a special case of Casey’s. We will define the tangents between circles and their properties, and then we will look at Casey’s theorem. Before proving the inverse of the theorem, we will also define what an inversion is and how it maps objects in a plane. We will conclude the thesis with a demonstration of some of the consequences.
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