Is it possible to construct the midpoint of a segment in the hyperbolic plane using the

set squareonly?

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 $O$.

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