Longuerre's theorem
In mathematics, particularly in Euclidean geometry, Longuerre's theorem is a result concerning the collinearity of points constructed from a cyclic quadrilateral. It is a generalization of the Simson line, which states that the three projections of a point on the circumcircle of a triangle to its sides are collinear.Sung Chul Bae, Young Joon Ahn (2012). "Envelope of the Wallace-Simson Lines with Signed Angle α". J. of the Chosun Natural Science. 5 (1): 38–41.
Statement
Longuerre's theorem. Let be a cyclic quadrilateral, and let be an arbitrary point. For each triple of vertices, construct the Simson line of with respect to that triangle. Let be the projection of onto the Simson line corresponding to the triangle formed by omitting vertex . Then the four points are collinear.Yu Zhihong (1996). "Proof of Longuerre's theorem and its extensions by the method of polar coordinates". Pacific Journal of Mathematics. 176 (2): 581–585.
Longuerre's theorem can be generalized to cyclic -gons.