classical geometry · circle inversion · tangency
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Around 200 BCE, Apollonius of Perga posed a question that would obsess geometers for two millennia: given three circles in the plane, find all circles simultaneously tangent to all three.
There are generally eight solutions — each corresponding to a different choice of internal or external tangency with each of the three given circles. The solutions come in four pairs related by circle inversion, the transformation that maps points inside a reference circle to points outside it.
The coaxial families view shows the two orthogonal pencils of circles passing through the two limiting points — a structure that appears throughout complex analysis and electrostatics.