Henry Segerman and I recently considered the following question:


Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of points which give rise to a hyperbolic triangle of the given area?


We found it a fun exercise in hyperbolic geometry to show that the answer is a Euclidean straight line, or an arc of a Euclidean circle. As this requires only elementary properties of hyperbolic geometry, we strongly suspect it should be known, but have thus far been unable to find a reference for it. Does anyone know whether it's known, and if so, where one can find it?



Locus of equal area hyperbolic triangles
Nice problem! After googling "hyperbolic triangles of equal area on a fixed base" I found the paper "Extremal properties of the principal Dirichlet eigenvalue for regular polygons in the hyperbolic plane" by Karp and Peyerimhoff. Their Theorem 7 looks like the result you ask for. In a footnote KP refer to a 1965 book of Fejes Toth, "Regulare Figuren", as containing the same result. There appears to be an English version "Regular Figures" published in 1964. 


This question seems very interesting in spherical geometry, as well.


