A hyperbolic concurrency theorem (Q1385214)

The authors received their motivation for this paper from the following Euclidean theorem discovered by \textit{L. Hoehn} [College Math. J. 22, No. 2, 129-132 (1997)]: ``Suppose that the (extended) sides of \(\Delta ABC\) intersect a given circle in six points, namely \(\alpha\), \(\alpha'\) on \(BC\), \(\beta\), \(\beta'\) on \(CA\), and \(\gamma\), \(\gamma'\) on \(AB\). If \(A^*= B\beta'\cap C\gamma\), \(B^*= C\gamma'\cap A\alpha\), \(C^*= A\alpha'\cap B\beta\), then \(AA^*\), \(BB^*\), \(CC^*\) are concurrent or parallel.'' Since, the theorem involves incidence properties only, the authors state and prove it now as a projective theorem (which turned out to be well-known). Then, using the Beltrami model of a hyperbolic plane, the theorem is interpreted as a hyperbolic theorem where the six points labeled by Greek letters serve as the common points at infinity (``ends'') of the lines in a pencil of parallels. As in the Euclidean case, the proof is based on Ceva's theorem.