$[\quad$ Constructions (other). ] [ Congruent triangles. Criteria for congruence (other).]

On a plane, two lines and a point $M$ are given. Find a point $X$ on one of the lines such that the segment $M X$ is bisected by the other line.
Through point $M$ draw a line parallel to one of the given lines.

## Solution

Let the given lines $l$ and $m$ intersect at point $O$, and point $M$ does not lie on either of them (see left figure). Through point $M$, draw a line parallel to $l$. Let this line intersect line $m$ at point $A$. Then the desired point $X$ is the intersection of the line passing through the midpoint of segment $OA$ and point $M$, with line $l$. In this case, the problem has two solutions.

If point $M$ lies on one of the given lines, there are no solutions.

If the given lines are parallel (see right figure), the problem has either an infinite number of solutions (when point $M$ is at a distance from one of the lines that is twice the distance from the other), or no solutions (in all other cases).
![](https://cdn.mathpix.com/cropped/2024_05_06_9617f685748a8dfcd82ag-12.jpg?height=272&width=1276&top_left_y=1619&top_left_x=389)

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