SELECTION and TRAINING SESSION

Since the lines rotate uniformly the angles between the lines $a_1(t_1),a_1(t_2)$ and between the lines $a_2(t_1),a_2(t_2)$ are equal for any moments $t_1,t_2$. Hence the points $A_1,A_2,A(t_1),A(t_2)$ lie on the circle, i.e. all points $A(t)$ lie on the same circle. Moreover if the lines rotated by $\varphi$ then the point $A(t)$ passed along the arc with the measure $2\varphi$. Therefore the points $A(t)$, $B(t)$ and $C(t)$ are moving uniformly and with the same angular velocity along some three circles. And one full-circle rotation there were two moments $t_1$ and $t_2$ such that the triangles $A(t_1)B(t_1)C(t_1)$ and $A(t_2)B(t_2)C(t_2)$ were equilateral and equally oriented.

Consider this situation on the complex plane and without loss of generality let the points make one full-circle rotation during one unit of time. Let $o_1,o_2,o_3$ be the coordinates of the centers and $r_1,r_2,r_3$ be the radii of the corresponding circles. Then the points $A(t),B(t)$ and $C(t)$ have the coordinates $o_1+r_1{e}^{t+{\phi }_1}$, $o_2+r_2{e}^{t+{\phi }_2}$ and $o_3+r_3{e}^{t+{\phi }_3}$ where ${\phi }_1,{\phi }_2$ and ${\phi }_3$ are angles corresponding to the initial positions of points. Then we know that $$\frac{(o_2+r_2{e}^{t_1+{\phi }_2})-(o_1+r_1{e}^{t_1+{\phi }_1})}{(o_3+r_3{e}^{t_2+{\phi }_2})-(o_1+r_1{e}^{t_2+{\phi }_1})}=\frac{(o_2+r_2{e}^{t_2+{\phi }_2})-(o_1+r_1{e}^{t_2+{\phi }_1})}{(o_3+r_3{e}^{t_3+{\phi }_2})-(o_1+r_1{e}^{t_3+{\phi }_1})}=\xi ,$$ where $\xi$ is the third root of unity. Note that after simple transformations the equation $$\frac{(o_2+r_2{e}^{t+{\phi }_2})-(o_1+r_1{e}^{t+{\phi }_1})}{(o_3+r_3{e}^{t+{\phi }_3})-(o_1+r_1{e}^{t+{\phi }_1})}=\xi$$ reduces to a linear equation of the variable $e^t$. Since this equation has two different solutions $e^{t_1}$ and $e^{t_2}$ on a unit circle, it must be degenerate, i.e. it is an identity. Therefore the triangle $A(t)B(t)C(t)$ is always equilateral.