Let $T$ be the set of all plane triangles. The problem is to find $t \in T$ s.t.h. a predicate $P(t)$ holds.
At present, I'm doing this by a form of randomized search procedure (effectively via a transformation $m: T \rightarrow T$, which implicitly defines a search graph).
However, it is known that $P(T)$ is invariant under rotation and translation.
I'm aware that the Erlangen program was famously concerned with the connection between geometry and group theory, so my question is:
How might one express the search space so that it can be `quotiented' by the symmetries implied by the above?
Use the SSS Theorem: a plane triangle is determined, up to rotation and translation, by the lengths of its three sides written in nondecreasing order. So the quotient is identified with ordered triples $(a,b,c)$ such that $0 < a \le b \le c$ and such that the triangle inequality $c \le a+b$ holds.