I've edited Steve Gray's message to kill the PDF attachment. You can get the paper directly from him (stevebg@adelphia.net). Rich ----- Gene Salamin asked:

Could you provide a statement of the 100 year old result, and of your extention to it?

Steve Gray replied: The 98-year-old result has been called the Douglas-Neumann theorem, but was first found by K. Petr in 1905, so I call it the Petr-Douglas-Neumann theorem, after Jesse Douglas and B.H. Neumann. It concerns layers of isosceles triangles constructed on the sides of an n-gon. After n-2 layers have been constructed, the resulting n-gon is regular, and after one more layer, a single point results. The first layer of triangles have apex angle 360/n, the second twice that, the third 3x360/n, etc. In my generalization the triangles used are not isosceles but are derived from an arbitrary-shaped "model" n-gon which is in general not similar to the one the triangles are placed on. After n-2 layers one gets an n-gon similar to the model, and after one more layer, a single point results. This is the first of several generalizations I have found and proved. I am currently working on another generalization which, if I ever find and prove it, will be the basis of another paper. However I already have enough material for at least one more paper, and probably two or three. One interesting thing about all these theorems is that they are "angle-only" theorems, that is, the constructions and results can be stated in natural fashion without reference to distances or lengths, which appear only secondarily. They are among the few theorems involving n-gons, where n is arbitrary. In some of my variations n must be prime, and in some others, certain of the construction parameters must have (simple) number-theoretic relationships. I'm enclosing a PDF of the final paper. If you have any more specific questions, please write back, although I do want to keep certain details of the other theorems private until I send in the next paper. Coincidentally, the same issue of the Monthly contains a geometry problem I submitted in 9/2001 that I had forgotten about. Steve Gray