Singular Fr\'egier Conics in Non-Euclidean Geometry

The hypotenuses of all right triangles inscribed into a fixed conic $C$ with fixed right-angle vertex $p$ are incident with the Fr\'egier point $f$ to $p$ and $C$. As $p$ varies on the conic, the locus of the Fr\'egier point is, in general, a conic as well. We study conics $C$ whose Fr\'egier locus is singular in Euclidean, elliptic and hyperbolic geometry. The richest variety of conics with this property is obtained in hyperbolic plane while in elliptic geometry only three families of conics have a singular Fr\'egier locus.