Fred wins his bet! I've just submitted a paper, ``Five-point circles, the 76-point sphere, and the Pavillet tetrahedron'' to the Monthly. The theme is that triangle geometry is not dead. I would attach a copy if attachments were allowed; I will entertain individual requests. Of course, I'm hoping that the paper will be accepted, and I also now hope to include a stop press paragraph with the following theorem: The external angle-bisectors of a triangle meet the opposite edges in collinear points. Call it the zero-sum line because: The sum of the directed distances to the edges of a triangle from points on its zero-sum line is zero. So go ahead and spoil my fun and tell me that that's well known to those who well know it. Certainly the first part is well known since it follows immediately from (the converse of) Menelaus's theorem, so the zero-sum line may already have some other name. The Gergonne line would be a good guess, but that's not quite right. Note that the zero-sum line (and the Gergonne line) are the line at infinity if the triangle is equilateral. Best, R. On Sat, 29 Oct 2011, Fred lunnon wrote:
Nice one, Dan --- bet that isn't in Clark Kimberling's list! WFL
On 10/28/11, Dan Asimov <dasimov@earthlink.net> wrote:
Given an arbitrary triangle T in R^2, characterize p in R^2 such that the sum of distances
S(p) := d(p, L_1) + d(p, L_2) + d(p, L_3)
is minimized, where the L_j are the affine lines containing the sides of T.
--Dan
"Things are seldom what they seem." --W.S. Gilbert _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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