Six-periodic isosceles-triangle orbit conjecture

Let $q_t$ and $r_t$ be the maps used to describe the two symmetry-related images of a triangle, and let a triangle be non-equilateral when it is not equilateral. Six-periodic isosceles-triangle orbit conjecture. There exist values $t_m\approx0.53$ and $t_M\approx0.57$ such that, for every $t\in(t_m,t_M)$, there exists a unique non-equilateral triangle ${\bf P}(t)$ for which $q_t({\bf P}(t))$ and $r_t({\bf P}(t))$ are both isosceles and have unit area. This conjecture is motivated by phase portraits showing an apparent six-periodic orbit; the equations defining the orbit were not solved analytically, but numerical solutions were found.