19.34. The midpoints of sides $B C$ and $B_{1} C_{1}$ of the regular 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 $B B_{1}$.
19.34. Let $M$ be the common midpoint of sides $B C$ and $B_{1} C_{1}, \boldsymbol{x}=\overrightarrow{M B}$ and $\boldsymbol{y}=\overrightarrow{M B_{1}}$. Let, further, $P$ be a rotational homothety with center $M$, rotation angle $90^{\circ}$, and coefficient $\sqrt{3}$, mapping point $B$ to $A$, and $B_{1}$ to $A_{1}$. Then $\overrightarrow{B B_{1}}=\boldsymbol{y}-\boldsymbol{x}$ and $\overrightarrow{A A_{1}}=P(\boldsymbol{y})-P(\boldsymbol{x})=P\left(\overrightarrow{B B_{1}}\right)$. Therefore, the angle between vectors $\overrightarrow{A A_{1}}$ and $\overrightarrow{B B_{1}}$ is $90^{\circ}$ and $A A_{1}: B B_{1}=\sqrt{3}$.