Open problems from antiquity solved with analytic geometry 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?
 A: I read this excerpt of Men of Mathematics as a private joke.
The very ordinary boy is Gauss of course. The number seventeen is a reference to the heptadecagon that Gauss constructed at age nineteen.
This is the first progress since the Greeks concerning compass and straight-edge construction of regular polygons. It makes use of cartesian geometry, arithmetics, trigonometry and requires the solving of a degree seventeenth polynomial equation. Here is the solution from the Disquisitiones Arithmeticae.
\begin{align} 16\,\cos\frac{2\pi}{17} = & -1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \\
                                                     & 2\sqrt{17+3\sqrt{17}-
                                                        \sqrt{34-2\sqrt{17}}-
                                                       2\sqrt{34+2\sqrt{17}}}\\
= & -1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \\
                                                     & 2\sqrt{17+3\sqrt{17}-
                                                        \sqrt{170+38\sqrt{17}}}.
 \end{align}
This would certainly have baffled the Greek. I understand childish in Bell's excerpt as refering to Gauss as a child prodigy. 
Another mathematician commonly endowed with supernatural powers as a child is Pascal, who supposedly rediscovered all the Euclid axioms at age eleven (debunked by his sister).
His treaty Essai sur les coniques written at age sixteen contains his 
famous hexagrammum mysticum theorem, which generalises a previous result by Pappus. This a result that would have certainly baffled the Greeks. 
Pascal used projective geometry to reduce the proof to the case of a circle. Nowadays there are many short elegant proofs of Pascal theorem using analytic geometry.
A: Provocative answer (but not too much): checking the Fifth Postulate is a triviality in the Cartesian plane.
A: 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.)
A: A classic problem in this category is Alhazen's billiard problem. I reproduce a quote from  100 Great Problems of Elementary Mathematics. The problem could not be solved using compass and ruler because its solution requires taking a cube root (see references at MathWorld).

A: This paper contains a very readable account of Descartes invention of analytic geometry and describes some questions that it can solve reasonably easily compared to methods familiar to the ancient greeks.
Even defining conics sections in the plane is quite clumsy using the directrix, focus and eccentricity. The analytic form of a second degree equation is easier to remember and clearly superior when computing actual points of intersection for example. 
Descartes' Theorem is one that is tough to prove or perhaps even discover without analytic geometry. Descartes himself used this as an example of the power of his method in his correspondence with Princess Elizabeth of Bohemia:
Bos, Erik-Jan, Princess Elizabeth of Bohemia and Descartes’ letters (1650-1665), Hist. Math. 37, No. 3, 485-502 (2010). ZBL1200.01012.
