Question 89
The lines $a_1, a_2, b_1, b_2, c_1, c_2$ passing, respectively, through the points $A_1, A_2, B_1, B_2, C_1, C_2$ rotate uniformly and with the same angular velocity about the corresponding points. At an arbitrary moment $t$ by $A(t)$ denote the intersection point of the lines $a_1$ and $a_2$. Points $B(t)$ and $C(t)$ are defined similarly. It turned out that within one $180^\circ$ 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.
Prove that the triangle $A(t)B(t)C(t)$ is always equilateral.
(Aliaksei Vaidzelevich)
Prove that the triangle $A(t)B(t)C(t)$ is always equilateral.
(Aliaksei Vaidzelevich)