Given any cyclic (inscribed) quadrilateral A1B1C1D1
For any point P on line C1D1, and any point Q on A1B1 :
Draw parallel (g1) to A1B1 from P, and parallel (g2) to C1D1 from Q.
Line PA1 intersects (g2) in B2, line PB1 intersects (g2) in A2
Line QC1 intersects (g1 in D2, line QD1 intersects (g1) in C2
Property (to be proved) :
Quadrilateral A2B2C2D2 is similar to quadrilateral A1B1C1D1, (hence is cyclic).
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Points A1,B1,C1,D1 can be moved on the circle, the quadrilateral needs not to be convex.
P and Q can be moved on the full lines C1D1 and A1B1
The property is generally false if A1B1C1D1 is not a cyclic quadrilateral.