42. Given three lines $l_{1}, l_{2}$ and $l_{3}$ and three points $A, B$ and $C$-one point on each of the lines. Draw a line $m$ intersecting lines $l_{1}, l_{2}$ and $l_{3}$ at points $X, Y$ and $Z$ such that $A X=B Y=C Z$.
42. Let the lines $l_{1}, l_{2}$ and $l_{3}$ not all be parallel to each other, for example, $l_{3}$ is not parallel to either $l_{1}$ or $l_{2}$. Suppose the problem is solved (Fig. 155). By Theorem 1, there exists

![](https://cdn.mathpix.com/cropped/2024_05_21_42e14062cf36c74a4d8dg-194.jpg?height=442&width=739&top_left_y=690&top_left_x=195)

Fig. 155.

a rotation that maps $A X$ to $C Z$, and a rotation that maps $B Y$ to $C Z$; the angles of rotation $\alpha_{1}$ and $\alpha_{2}$ are equal to the angles between $l_{1}$ and $l_{3}$, and $l_{2}$ and $l_{3}$, respectively, and the centers of rotation $O_{1}$ and $O_{2}$ are located as in the first solution of problem 41 a) - r). From the isosceles triangles $O_{1} X Z$ and $O_{2} Y Z$ with angles at the vertices $O_{1}$ and $O_{2}$, respectively, equal to $\alpha_{1}$ and $\alpha_{2}$, we find $\angle O_{1} Z X = 90^{\circ} - \frac{\alpha_{1}}{2}$, $\angle O_{2} Z Y = 90^{\circ} - \frac{\alpha_{2}}{2}$. Therefore, $\angle O_{1} Z O_{2} = \frac{\alpha_{1} \pm \alpha_{2}}{2}$, and thus, the point $Z$ can be found as the intersection of the line $l_{3}$ with the arc of the circle subtended by the segment $O_{1} O_{2}$, containing the known angle $\frac{\alpha_{1} + \alpha_{2}}{2}$ or $\frac{\alpha_{1} - \alpha_{2}}{2}$.

I. M. Yaglom

Each of the angles of rotation $\alpha_{1}$ and $\alpha_{2}$, and the corresponding centers of rotation $O_{1}$ and $O_{2}$, can be determined in two different ways (compare with the solution of the previous problems). Therefore, the maximum possible number of solutions to the problem is 16.

If $l_{1} \| l_{2} \| l_{3}$, then the problem has three solutions in the case where the points $A, B$, and $C$ do not lie on the same line (the solutions are given by the medians of the triangle $ABC$), and infinitely many solutions when $A B C$ is a straight line.