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In http://debart.pagesperso-orange.fr/seconde/contruc_cercle.html there are ten problems of contact concerning circles which are solved in a quite interesting way. The text is in French and the problem $10$ is the Apollonius’s consisting in finding a circle tangent to three given circles of which eight solutions are given.
Here a translation of the “Historique” of the subject.
“ The problem of the circle tangent to three circles is one of the great problems of history of geometry.
It was introduced by Pappus as the tenth and most difficult of the Treaty of contacts, one of the lost works of Apollonius.
In 1596, Adrien Romain (Van Roomen, Latinized as Adrianus Romanus, Flemish mathematician 1561-1615) propose a solution using a hyperbola, that Vieta considers not in accordance with the method of Ancients.
Indeed, the emergence of algebra in geometry allows him to claim that it is a problem of the second degree, so a plan problem which can be solve with "the ruler and compass."
Vieta published his own solution in 1600, in his Apollonius Gallus where he presents some lemmas for handling the similarities and where he exhibited the first nine situations listed above. It recognizes that the solutions to this problem tenth depend on the relative position of the three circles, but ignores the discussion of a lot of solutions and will have to wait until Descartes (1637) for discussion of particular solutions.
Until the nineteenth century, this problem will be one of the places of confrontation between the synthetic geometry (pure geometry) and analytical geometry."