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