A porism for cyclic quadrilaterals, butterfly theorems, and hyperbolic geometry

If there exists a cyclic quadrilateral whose sides go through the given four collinear points, then there are infinitely many such quadrilaterals inscribed in the same circle. We give two proofs of this porism; one based on cross-ratios, the other on compositions of hyperbolic isometries.Comment: 9 pages, 10 figures, to appear in Amer. Math. Monthl