Given any cyclic (inscribed) quadrilateral A_{1}B_{1}C_{1}D_{1}

For any point P on line C_{1}D_{1}, and any point Q on A_{1}B_{1} :

Draw parallel (g1) to A_{1}B_{1} from P, and parallel (g2) to C_{1}D_{1} from Q.

Line PA_{1} intersects (g2) in B_{2}, line PB_{1} intersects (g2) in A_{2}

Line QC_{1} intersects (g1 in D_{2}, line QD_{1} intersects (g1) in C_{2}

Property (to be proved) :

Quadrilateral A_{2}B_{2}C_{2}D_{2} is similar to quadrilateral A_{1}B_{1}C_{1}D_{1},
(hence is cyclic).

Points A_{1},B_{1},C_{1},D_{1} can be moved on the circle, the quadrilateral needs not to be convex.

P and Q can be moved on the full lines C_{1}D_{1} and A_{1}B_{1}

The property is generally false if A_{1}B_{1}C_{1}D_{1} is not a cyclic quadrilateral.