Is it possible to construct the midpoint of a segment in the hyperbolic plane using the set square only?

With the set square one can

• draw the line through the given two points and
• drop the perpendicular from the given point to the given line.

Is it possible to construct the midpoint of a segment in the hyperbolic plane using the set square only?

With the set square one can

• draw the line through the given two points and
• drop the perpendicular from the given point to the given line.

The following construction produce the point $X'$ which is centrally symmetric to the point $X$ with respect to point $O$.

1. Draw line $(OX)$ and let $m$ be the line perpendicular to $(OX)$ through $O$.
2. Draw yet two perpendicular lines $l$ and $l'$ through $O$.
3. Find the foot point $Y$ of $X$ on $l$.
4. Draw the line through $Y$ perpendicular to $m$ and let $Z$ be its intersection with $l'$.
5. Finally, $X'$ is the footpoint of $Z$ on $l$.