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Paper Perplexities
Jan. 31st, 2016 09:14 amI'm starting to flesh out the first paper in my proposed trilogy, Towards a Taxonomy of Polygons. Unfortunately, the new version of Scientific Workplace - the specialized word processor I use - is significantly different from the previous, and I'm having a little learning-curve trouble. (If I understand correctly, at least one of the changes is a major improvement, but it involves the late stages, not the initial organization.)
The second half of the first paper will discuss classes of triangles (and I've recently stumbled on some new material for that). I'm not sure how much of it is actually new, so I've ordered a copy of John Conway's The Triangle Book, which should enlighten me on a number of things. Unfortunately, the book isn't coming out until the beginning of April. :grmph: I thought it had been published a couple of years ago.
Still, I'm pleased with the new stuff. The thing is this: every triangle has certain interesting lines associated with it; the best known is the Euler line, which passes through the centroid, the circumcenter, and the orthocenter (as well as a bunch of other interesting points). I discovered a while back that each such line is associated with a pair of classes of triangles, characterized by (in one case) the line being parallel to an edge and (in the other) its being parallel to a median. What I didn't realize until this past week is that this can happen in two different ways: a class might be associated with the parallel-side case for one line and with the parallel-median for another. As a bonus, the perpendicular cases also pop up, although less nicely than the parallel cases. The neatest bit is this. The four most important lines associated with a triangle are the Euler line, the Brocard axis, the symmedian trail, and the orthic axis. The very first pair of triangle classes that I discovered, the isosceles and isomedial classes, can be characterized by each of these except the Brocard axis. Which is neat.
(I'm not claiming to have discovered isosceles triangles; I do think I discovered isomedial triangles and the way the class pairs with the class of isosceles triangles.)
The second half of the first paper will discuss classes of triangles (and I've recently stumbled on some new material for that). I'm not sure how much of it is actually new, so I've ordered a copy of John Conway's The Triangle Book, which should enlighten me on a number of things. Unfortunately, the book isn't coming out until the beginning of April. :grmph: I thought it had been published a couple of years ago.
Still, I'm pleased with the new stuff. The thing is this: every triangle has certain interesting lines associated with it; the best known is the Euler line, which passes through the centroid, the circumcenter, and the orthocenter (as well as a bunch of other interesting points). I discovered a while back that each such line is associated with a pair of classes of triangles, characterized by (in one case) the line being parallel to an edge and (in the other) its being parallel to a median. What I didn't realize until this past week is that this can happen in two different ways: a class might be associated with the parallel-side case for one line and with the parallel-median for another. As a bonus, the perpendicular cases also pop up, although less nicely than the parallel cases. The neatest bit is this. The four most important lines associated with a triangle are the Euler line, the Brocard axis, the symmedian trail, and the orthic axis. The very first pair of triangle classes that I discovered, the isosceles and isomedial classes, can be characterized by each of these except the Brocard axis. Which is neat.
(I'm not claiming to have discovered isosceles triangles; I do think I discovered isomedial triangles and the way the class pairs with the class of isosceles triangles.)
