  There has been recent work using Shape Theory to answer the longstanding and
conceptually interesting problem of what is the probability that a triangle is
obtuse. This is resolved by three kissing cap-circles of rightness being
realized on the shape sphere; integrating up the interiors of these caps
readily yields the answer to be 3/4. We now generalize this approach by viewing
rightness as a particular value of maximal angle, and then covering the shape
sphere with the maximal angle flow. Therein, we discover that the kissing
cap-circles of rightness constitute a separatrix. The two qualitatively
different regimes of behaviour thus separated both moreover carry distinct
analytic pathologies: cusps versus excluded limit points. The equilateral
triangles are centres in this flow, whereas the kissing points themselves --
binary collision shapes -- are more interesting and elaborate critical points.
The other curves' formulae and associated area integrals are more complicated,
and yet remain evaluable. As a particular example, we evaluate the probability
that a triangle is Fermat-acute, meaning that its Fermat point is nontrivially
located; the critical maximal angle in this case is 120 degrees.
