I thinks $\alpha_i$ can greater $\pi$

In $n=3$, let $A_1, A_2, A_3$ are points $A, B, C$; $B_1, B_2, B_3$ are Fermat triangle of $ABC$. $D_1=D_2=D_3=P$ is the arbitrary point in the plane of $ABC$; $C_1, C_2, C_3$ are the reflection of $A$, $B$, $C$ in $P$, $\ell=1/2$. This is one case of generalization of Dao-Nhi equilateral triangle

In $n=3$, let $A_1, A_2, A_3$ are points $A, B, C$; $B_1, B_2, B_3$ are Fermat triangle of $ABC$. $D_1=D_2=D_3=$ the centroid of $ABC$; $C_1, C_2, C_3$ are the Midpoint of $BC, CA, AB$ $\ell=1/3$ we get Napoleon theorem