The tangency problem of Apollonius: three looks.

During the great period of Greek geometry, an intriguing construction challenge was posed by Apollonius. Some eighteen centuries later, there was revived interest in the problem, and it continues today. François Viète, Isaac Newton, and Joseph-Diaz Gergonne were among the many mathematicians who solved the problem. They lived in different eras and had very different approaches. Viète stayed close to geometric fundamentals, Newton used his grasp of conics, and Gergonne employed a new understanding of inversion geometry properties. [ABSTRACT FROM AUTHOR]