Szilasi, Zoltán2025-12-042025-12-042025-12-01Teaching Mathematics and Computer Science, Vol. 23 No. 2 (2025) , 195-2111589-7389https://hdl.handle.net/2437/399463In the Cayley–Klein model, we review some basic results concerning the geometry of hyperbolic triangles. We introduce a new definition of the circumcircle of a hyperbolic triangle, guaranteed to exist in every case, and describe its main properties. Our central theorem establishes, by means of purely elementary projective geometric arguments, that a hyperbolic triangle has a nine-point conic if and only if it is a right triangle. Subject Classification: 51M09application/pdfCayley-Klein planehyperbolic trianglesFeuerbach circleeleven-point conicfolyóiratcikkOpen AccessZoltán Szilasihttps://doi.org/10.5485/TMCS.2025.15646Teaching Mathematics and Computer Science223Teach. Math. Comp. Sci.2676-8364