Let the three vertices of the triangle be A,B,C, so AB is the middle S and BC is the end S. Constructing the triangle given the three parameters involves drawing AB, drawing a line from A at the specified angle (where there are two choices--on the left or on the right), and drawing a circle at B of length BC (which intersects the line from A at two points, resulting in two ABC triangles). The circumcircle of either ABC is the same, because the angle A is the same so BC is the same length. Is that any clearer? --ms On 06-Apr-15 17:51, James Propp wrote:
Thanks, but I'm not seeing this. Can you tell me how to construct your diagram step by step, since the one I'm drawing is apparently different in important respects and doesn't show me what you're seeing?
Jim Propp
On Mon, Apr 6, 2015 at 4:40 PM, Mike Speciner <ms@alum.mit.edu> wrote:
If the middle S is a chord of a circle, then the end S can be a chord going either way on that circle and the angle A will be equal either way. So I'm guessing X is the circumcircle.
--ms
On 06-Apr-15 15:51, James Propp wrote:
Back about 30-40 years ago, I posed a problem in Mathematics Magazine (or maybe the Monthly) asking whether any triangle could be linked to a non-congruent similar triangle via a sequence of triangles, each "SSA-congruent" to the one before and the one after. There was a cute solution that pointed out that this is impossible because two SSA-congruent triangles have the same X, where X was some triangle statistic (like perimeter, inradius, or circumradius, but slightly less well-known) that scales linearly under similarity.
Can anyone (a) figure out what X was, or (b) locate my problem and the solution?
Jim Propp
PS: In my original submission I proposed the term "ASS-congruent", which struck me as both more pronounceable and more apt, but the stodgy problems editor who reigned at the time deemed this too vulgar. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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