An Amazing Theorem about a General Quadrilateral
By Shalosh B. Ekhad
For any two points,P,Q let, Line(P,Q) be the line joining P and Q
For any two lines L and N let, Point(L,N) be the point of intersection of L and N
Theorem: Let, P[1], P[2], P[3], P[4], be ARBITRARY four points on the plane
The following is true
the following three points are COLLINEAR
Point(Line(P[1], P[2]), Line(Point(Line(P[1], P[3]), Line(P[2], P[4])), Point(Line(P[1], P[4]), Line(P[2], P[3]))))
Point(Line(P[1], P[3]), Line(Point(Line(P[1], P[2]), Line(P[3], P[4])), Point(Line(P[1], P[4]), Line(P[2], P[3]))))
Point(Line(P[2], P[3]), Line(Point(Line(P[1], P[2]), Line(P[3], P[4])), Point(Line(P[1], P[3]), Line(P[2], P[4]))))
This ends this article, that took, 0.037, seconds to produce.
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An Amazing Theorem about a General Pentagon
By Shalosh B. Ekhad
For any two points,P,Q let, Line(P,Q) be the line joining P and Q
For any two lines L and N let, Point(L,N) be the point of intersection of L and N
Theorem:
Let, P[1], P[2], P[3], P[4], P[5], be ARBITRARY five points on the plane
The following is true
the following three lines are Concurrent
Line(P[1], P[2])
Line(Point(Line(P[1], P[3]), Line(P[4], P[5])), Point(Line(P[2], P[4]), Line(P[3], P[5])))
Line(Point(Line(P[1], P[4]), Line(P[3], P[5])), Point(Line(P[2], P[3]), Line(P[4], P[5])))
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also the following three lines are Concurrent
Line(P[1], Point(Line(P[2], P[3]), Line(P[4], P[5])))
Line(P[2], Point(Line(P[1], P[4]), Line(P[3], P[5])))
Line(P[5], Point(Line(P[1], P[3]), Line(P[2], P[4])))
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also the following three lines are Concurrent
Line(P[1], Point(Line(P[2], P[3]), Line(P[4], P[5])))
Line(Point(Line(P[1], P[2]), Line(P[3], P[4])), Point(Line(P[1], P[3]), Line(P[2], P[5])))
Line(Point(Line(P[1], P[4]), Line(P[2], P[5])), Point(Line(P[1], P[5]), Line(P[3], P[4])))
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also the following three lines are Concurrent
Line(Point(Line(P[1], P[2]), Line(P[3], P[4])), Point(Line(P[1], P[3]), Line(P[2], P[4])))
Line(Point(Line(P[1], P[2]), Line(P[3], P[5])), Point(Line(P[1], P[3]), Line(P[2], P[5])))
Line(Point(Line(P[2], P[4]), Line(P[3], P[5])), Point(Line(P[2], P[5]), Line(P[3], P[4])))
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This ends this article, that took, 5.938, seconds to produce.
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An Amazing Theorem about an Inscribed Hexagon in Conic
By Shalosh B. Ekhad
For any two points,P,Q let, Line(P,Q) be the line joining P and Q
For any two lines L and N let, Point(L,N) be the point of intersection of L and N
Theorem: Let, P[1], P[2], P[3], P[4], P[5], P[6], be ARBITRARY six points lying on a conic section
The following is true
the following three points are COLLINEAR
Point(Line(P[1], P[2]), Line(P[3], P[4]))
Point(Line(P[1], P[5]), Line(P[3], P[6]))
Point(Line(P[2], P[6]), Line(P[4], P[5]))
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This ends this article, that took, 0.501, seconds to produce.
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This took, 6.502, seconds