I'm pretty sure it is possible to construct a triangle in the Klein model of hyperbolic geometry such that the perpendicular bisectors are divergently parallel, but I'm struggling to do so. I've been trying to do the problem just by drawing it out and not wanting to calculate distances or anything. Any advice on how I would want to choose the lines that construct my triangle inside the circle?
2026-03-27 03:46:15.1774583175
How to construct a triangle with divergently parallel perpendicular bisectors?
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The construction you want is easy. In your belatedly cited source, see figure 3.1. $AB$ is divergently parallel to $CD$. Construct a short segment perpendicularly bisected by $AB$ (easiest near $A$) that intersects a short segment perpendicularly bisected by $CD$. The add the third segment. (Note that both short segments must lie entirely in the circle (except possibly for one point on the circle).)
Edit: Deleted correct but irrelevant claim about altitudes due to mentally transposing "perpendicular bisector" and "altitude".