TITLE: Open problems from antiquity solved with analytic geometry
QUESTION [34 upvotes]: E. T. Bell wrote in Men of Mathematics:
Though the idea behind it all is childishly simple, yet the method of analytic geometry is so powerful that very ordinary boys of seventeen can use it to prove
results which would have baffled the greatest of the Greek geometers --
Euclid, Archimedes, and Apollonius. 
I don't necessarily believe everything in the book, but this passage sounds plausible enough to make me wonder.
Are there any good examples of an open problem from antiquity which seemed inaccessible, but was later easily solved by converting it to the Cartesian plane?

REPLY [2 votes]: I come across what looks like a good fit in Brianchon, Solution de plusieurs problèmes de géométrie, J. École Polytechnique 4, nº 10 (1810) 1–15, page 5:

Pappus reports also that the Greek geometers had tried in vain to solve this more general problem.
« Given a circle and three poles, arranged in arbitrary manner, inscribe in this circle a triangle whose sides, extended if necessary, each go through one of the given poles. »
With the help of analysis applied to geometry, the moderns easily overcame the difficulty, and this once famous question now amounts to very little; Lagrange has given a beautiful analytic solution (Mémoire de Berlin, 1776)

(etc.; according to Senapati (2019) Pappus had solved the case where the 3 points are aligned, in Mathematicae collectiones, Book 7, Prop. 117.)