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. 
Ivan Izmestiev
math.MG