Some progress(?): WLOG let $A,B,C$ be in counterclockwise order and have the square on $AB$, the regular $m$-gon on $BC$, and the regular $n$-gon on $CA$. If we identify the plane with $\mathbb{C}$ and work with complex numbers, then setting $a,b,c$ as the vertices of the triangle, $\zeta_m = e^{2\pi i/m}, \zeta_n = e^{2\pi i/n}$, and $\omega=e^{\pi i/3}$, the given conditions are equivalent to \[\displaystyle \frac{b-c}{b-a} + i\cdot\frac{c-a}{b-a} = \frac{\omega + i\zeta_n}{\omega\cdot\zeta_m + 1} = \frac{\cos\left(\pi/n + \pi/12\right)}{\cos\left(\pi/m + \pi/6\right)}\cdot e^{\pi i(2/3 +1/n+1/m)}.\] Does anyone have any ideas about how one could proceed from here?