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