**Eugen J. Ionaşcu,
The ``Circle" of Apollonius in Hyperbolic Geometry,**

**Forum Geometricorum,
18 (2018) 135—140.**

Abstract.
In Euclidean geometry the circle of Apollonius is the locus of points in the
plane from which two collinear adjacent segments are perceived as having the
same length. In Hyperbolic geometry, the analog of this locus is an algebraic
curve of degree four which can be bounded or “unbounded”. We study this locus and
give a simple description of this curve using the Poincaré
half-plane model. In the end, we give the motivation of our investigation and
calculate the probability that three collinear adjacent segments can be seen as of the same positive length under some
natural assumptions about the setting of the randomness considered.

Return to Forum Geom., 18
(2018) Table of Contents