mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, particularly in

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, Longuerre's theorem is a result concerning the

collinearity In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...

of points constructed from a

cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...

. It is a generalization of the

Simson line In geometry, given a triangle and a Point (geometry), point on its circumcircle, the three closest points to on lines , , and are collinear. The line through these points is the Simson line of , named for Robert Simson. The concept was first ...

, 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 $A_1A_2A_3A_4$ be a
cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...
, and let $P$ be an arbitrary point. For each triple of vertices, construct the
Simson line In geometry, given a triangle and a Point (geometry), point on its circumcircle, the three closest points to on lines , , and are collinear. The line through these points is the Simson line of , named for Robert Simson. The concept was first ...
of $P$ with respect to that triangle. Let $D_i$ be the
projection Projection or projections may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphics, and carto ...
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 In geometry, collinearity of a set of Point (geometry), points is the property of their lying on a single Line (geometry), line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, t ...
.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 Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...

n

-gons.

References

{{reflist Euclidean geometry Theorems about quadrilaterals

Statement

See also

References