## Apollonian Circles

In 200 BC, Apollonius of Perga posed the following question: “Using only a straightedge and compass, construct the circle or circles tangent to three given circles.” Scores of history’s most famous mathematicians have solved this problem in as many ways. I recreated Joseph Gergonne’s 19th century solution, filling in small proofs of my own to justify steps he made without explicit justification.

The solutions to Apollonius’ problem have been applied to a number of applications in recent history, including navigation systems and disk packing problems. Additionally, if the three circles in Appolonius’ problem are mutually tangent, the fourth circle can be drawn iteratively, yielding fractal designs like the one shown.