Given A=(a,0), B=(0,0), and C=(0,a)
Let f(a)=the total number of unit equilateral triangles XYZ that can be formed such that the lengths AX, BY, and CZ are all 1 unit.
Give a piecewise definition by intervals for f(a)
(In reply to
re: more precision by Charlie)
BTW, for example, the likely candidate for a spurious (12th) solution for a=0.73486, would be Z near (2.24457,0.79087), Y near (2.31,0.206), X near (1.41,0.23). Just as Y approaches being 1 unit from B (as Z is moved around the circle about C), from both sides, as the X's are getting closer together also, Z gets too far from A to have both ZX and XA be equal to 1.
For 1.956, the one near X at (0.16,0.55), Y at (0.84,0.55) and Z at (0.34,1.42) is spurious, though there is another one very nearby which is real, and combined with two others (X near (0.26,0.11),Y near (0.41,0.09),Z near (0.48,1.09), but offset in two directions slightly) make 3.

Posted by Charlie
on 20090915 00:30:43 