A non-Miquelian Laguerre plane satisfying a theorem of Miquelian type (Q1385212)

It was shown by \textit{R. Löwen} and \textit{U. Pfüller} [Geom. Dedicata 23, 73-85 (1987; Zbl 0615.51006)] that with each differentiable convex function \(f: {\mathbb R} \to {\mathbb R}\) whose derivative is a bijection one can associate a certain two-dimensional locally compact Laguerre plane. The author shows that these Laguerre planes satisfy the following theorem of Miquelian type. Let \(A_1,\dots,A_4\) and \(B_1,\dots, B_4\) be two quadruples of concyclic points such that \(A_i\) is parallel to \(B_i\) for \(i = 1,\dots,4\). If \(A_1, B_2, B_3\) and \(A_4\) are concyclic then so are \(B_1, A_2, A_3\) and \(B_4\). The author considers only the special case \(f(x) = x^4\) but his proof also works for general \(f\).