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51N: Analytic and descriptive geometry

Introduction

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").

Subfields

• 51N10: Affine analytic geometry
• 51N15: Projective analytic geometry
• 51N20: Euclidean analytic geometry
• 51N25: Analytic geometry with other transformation groups
• 51N99: None of the above but in this section

Parent field: 51: Geometry

Browse all (old) classifications for this area at the AMS.

Selected topics at this site

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.

• First, a statement of the problem (the original post is lost, but was unclear anyway).
• I had played around with the problem a bit; for example, here is some maple code seeking triangles.
• I made an initial post summarizing the condition for existence and the infinitude of solutions in the positive cases. A response is attached (which was more on-the-mark than I thought at the time).
• Meanwhile, I did some reading on the problem. Since I was at the time teaching a course on elliptic curves, when I found the relation of this problem to that topic I subjected my students to it. Here are some course notes clarifying details of the relationship.
• For the benefit of the USENET community I posted a summary of the whole matter in a lengthy second post.
• This prompted an email exchange with the original poster in which he outlines a geometric (albeit not "Euclidean") proof of Chapple's identity.