How to solve something like this? Consider the problem of covering the largest possible equilateral triangle with two unit squares.
I make a sketch in Geometer's Sketchpad first.
Then put the equations right into Mathematica. z0==z1==1.
Root[144 + 96 #1 + 16 #1^2 + 72 #1^3 - 168 #1^4 + 9 #1^6 &, 3]
About 1.322159389397273286325082341278424475
