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)
I had not finished solving this when I submitted it. It seems to be harder than I thought to get the exact solutions since the equations appear to become fourth degree polynomials and the algebra is easy to mess up.
I used geometers sketch pad as well, not considering its limits of accuracy. The points you are arguing, as I found them:
Near .70 my best guess is that this is sqrt(1/2) and gives 8 solutions. In fact the interval (.63507,sqrt(1/2)] gives 8 and then it jumps directly to 12. This is the only jump I noticed without a half-step and a tangency.
The point with 2 solutions is at about 1.95755 by my reckoning.
Posted by Jer
on 2009-09-15 10:25:18