[ $\underline{\text { Rotational Homothety }}$ ]

The midpoints of sides $B C$ and $B_{1} C_{1}$ of the equilateral triangles $A B C$ and $A_{1} B_{1} C_{1}$ coincide (the vertices of both triangles are listed in a clockwise direction). Find the measure of the angle between the lines $A A_{1}$ and $B B_{1}$, as well as the ratio of the lengths of segments $A A_{1}$ and $B B_{1}$.