[ Midline of a trapezoid ]

[ GML - a line or a segment ]

Find the geometric locus of the midpoints of segments with endpoints on two given parallel lines.

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Let $l_{1}$ and $l_{2}$ be given parallel lines, point $A$ lies on $l_{1}$, and point $B$ lies on $l_{2}$. Through the midpoint $M$ of segment $AB$, draw a line $l$ parallel to the given lines.

If $X$ and $Y$ are arbitrary points lying on lines $l_{1}$ and $l_{2}$ respectively and different from $A$ and $B$, then line $l$ contains the midline of trapezoid $ABYX$, that is, it passes through the midpoint $N$ of segment $XY$.

If point $X$ coincides with $A$ or $Y$ coincides with $B$, this is also true.

Conversely, any point $N$ on line $l$ is the midpoint of some segment with endpoints on lines $l_{1}$ and $l_{2}$ respectively. Indeed, let line $AN$ intersect $l_{2}$ at some point $Z$, then $MN$ is the midline of triangle $BAZ$.

## Answer

The line equidistant from the two given lines.