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.
Statement.
Longuerre's theorem. Let formula_1 be a cyclic quadrilateral, and let formula_2 be an arbitrary point. For each triple of vertices, construct the Simson line of formula_2 with respect to that triangle. Let formula_4 be the projection of formula_2 onto the Simson line corresponding to the triangle formed by omitting vertex formula_6. Then the four points formula_7 are collinear.
Longuerre's theorem can be generalized to cyclic formula_8-gons.