The Open Claim: The set of points in the box triangle whose barycentric coordinates are balanced is open.Proof: If {$(a,b,c)$} is balanced, then sufficiently slight changes in {$a$}, {$b$}, {$c$} will not change the topology of the path in the unfolding. Imagine the effect on the unfolding of making a slight increase, say, in {$b$}. Everything in the diagram moves. But the point at the upper right corner will move up and to the right at an angle of 45 degrees because {$(a,b,c)$} is balanced---every change to one side of the square is matched by an identical change to the other. The same is true for any (sufficiently slight) change in {$a$} or {$c$}. Thus the destination in the altered system remains the same. Note that the altered path will cross the same edges and in the same order. |