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. -------------------------------------------------------------------------------------------------------- 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]))) ---------------------------- 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]))) ---------------------------- 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]))) ---------------------------- 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]))) ---------------------------- This ends this article, that took, 5.938, seconds to produce. ------------------------------------------------------------------------------------------------------------------ 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])) -------------------------------- This ends this article, that took, 0.501, seconds to produce. ------------------------------ This took, 6.502, seconds