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. ^[1]

Statement

Longuerre's theorem. Let ${\displaystyle A_{1}A_{2}A_{3}A_{4}}$ be a cyclic quadrilateral, and let ${\displaystyle P}$ be an arbitrary point. For each triple of vertices, construct the Simson line of ${\displaystyle P}$ with respect to that triangle. Let ${\displaystyle D_{i}}$ be the projection of ${\displaystyle P}$ onto the Simson line corresponding to the triangle formed by omitting vertex ${\displaystyle A_{i}}$. Then the four points ${\displaystyle D_{1},D_{2},D_{3},D_{4}}$ are collinear. ^[2]

Longuerre's theorem can be generalized to cyclic ${\displaystyle n}$-gons. ^[2]

See also

• Polar coordinates
• Simson line

References

1. 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.
2. 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.