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.

Statement

Longuerre's theorem. Let $A_1{A}_2{A}_3{A}_4$ be a cyclic quadrilateral, and let $P$ be an arbitrary point. For each triple of vertices, construct the Simson line of $P$ with respect to that triangle. Let $D_i$ be the projection of $P$ onto the Simson line corresponding to the triangle formed by omitting vertex $A_i$. Then the four points $D_1,D_2,D_3,D_4$ are collinear.

Longuerre's theorem can be generalized to cyclic $n$-gons.

See also

• Polar coordinates
• Simson line