67. Given a line $l$, a point $A$ on it, and circles $S_{1}$ and $S_{2}$. Construct a triangle $A B C$ in which the line $l$ is the angle bisector of angle $A$, vertices $B$ and $C$ lie on circles $S_{1}$ and $S_{2}$ respectively, and the ratio of sides $A B$ and $A C$ is a given value $m: n$.
67. Let triangle $ABC$ be constructed (Fig. 180). Point $B$ is translated to point $C$ by a central similarity with center $A$, axis $l$, and similarity coefficient $\frac{n}{m}$. Therefore, point $C$ lies simultaneously on the circle $S_{\mathrm{z}}$ and on the circle $S_{1}^{\prime}$, obtained from $S_{1}$ by this central similarity (Fig. 180). Problem

![](https://cdn.mathpix.com/cropped/2024_05_21_42e14062cf36c74a4d8dg-227.jpg?height=398&width=451&top_left_y=422&top_left_x=178)

Fig. 180. can have two, one, or no solutions.

![](https://cdn.mathpix.com/cropped/2024_05_21_42e14062cf36c74a4d8dg-227.jpg?height=318&width=493&top_left_y=510&top_left_x=646)

Fig. 181.