- Congratulations! This post is now featured in our Staff Pick column as distinguished by a badge on your profile of a Featured Contributor! Thank you, keep it coming, and consider contributing your work to the The Notebook Archive!

Shenghui, thanks for your comment. I tried Butterfly Theorem.

Butterfly Theorem

Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD and BC intersect chord PQ at X and Y correspondingly. Then M is the midpoint of XY.

gs = GeometricScene[{"A", "B", "C", "D", "P", "Q", "M", "X", "Y"},
   {GeometricAssertion[CircleThrough[{"A", "P", "D", "B", "Q", "C"}], 
     "Counterclockwise"],
    GeometricAssertion[{{"A", "M", "B"}, {"C", "M", "D"}}, 
     "Collinear"],
    "M" == Midpoint[{"P", "Q"}],
    Line[{{"P", "X", "Q"}, {"A", "X", "D"}, {"P", "Y", "Q"}, {"C", 
       "Y", "B"}}],
    Line[{{"A", "B"}, {"C", "D"}, {"A", "D"}, {"B", "C"}}],
    Style[Line[{"X", "Y"}], Orange]
    }
   ];
RandomInstance[gs]
FindGeometricConjectures[gs]["Conclusions"]