51N: Analytic and descriptive geometry
Mostly this area includes topics related to ordinary analytic geometry as studied in secondary school. (Also included here is a rather lengthy analysis of a result known as "Poncelet's Porism").
Parent field: 51: Geometry
Browse all (old) classifications for this area at the AMS.
A compendium of plane curves
Here in addition is a collection of files on what I found to be a fascinating question in plane geometry. The following question was posted to sci.math: The problem, posed by Zeljko.Vrba@oleh.srce.hr (Zeljko Vrba), assumed two circles given in the plane, and asked for a characterization of the triangles which have these two circles as their in- and circum-scribed circles. I followed up with the observation that for most pairs of circles there are no such triangles, while otherwise there will be infinitely many; the cases are distinguished by a certain relation holding on the radii and the separation of the centers. (If the inner and outer radii are r and R respectively and if the distance between the centers is a then the existence of such a triangle implies a^2=R^2-2rR.)
It turns out that the condition ("Chapple's identity") for a triangle to exist is old, and while there are geometric/trigonometric proofs, I observe that the condition can be stated that certain line segments are congruent, yet I have no Euclidean proof of that fact. (Undoubtedly this is made harder by the non-uniqueness of the triangles, but this does suggest a question: if a true statement can be expressed in Euclidean terms, does it have a Euclidean proof?)
The existence of infinitely many solutions is "Poncelet's theorem" ("Poncelet's Porism") and is connected to some statements about elliptic curves.
Here are some files related to this topic.