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Famous computable theorems of geometry

POSTED BY: Kotaro Okazaki
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POSTED BY: EDITORIAL BOARD

I like these synthetic geometric features as well. These are powerful tools to explore Johnson's "Advanced Euclidean Geometry", Coolidge's "A treaties on the triangle and the sphere" and "the Heroic age of geometry".

Also would you like to try these functions to test Butterfly Theorem?
POSTED BY: Shenghui Yang

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"]

enter image description here enter image description here
POSTED BY: Kotaro Okazaki

Your code floats like a butterfly! For conjectures if you are only interested those related to point M, you can extract the statement by adding a Pattern into the FindGeometricConjectures: 

In[1]:= FindGeometricConjectures[gs,"M"==_]["Conclusions"]
Out[1]= {M==Midpoint[{Y,X}]}

or using HoldPattern["M" == _] to make the code more robust.
POSTED BY: Shenghui Yang
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