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

the Newton line is the line that connects the midpoints of the two

diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...

s in a

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

with at most two

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...

sides.Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. MAA, 2010, , pp. 108–109 ()

Properties

The

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...

s and that connect the midpoints of opposite sides (the

bimedian In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

s) of a convex quadrilateral intersect in a point that lies on the Newton line. This point bisects the line segment that connects the diagonal midpoints. By

Anne's theorem In Euclidean geometry, Anne's theorem, named after the French mathematician Pierre-Leon Anne (1806–1850) describes an equality of certain areas within a convex quadrilateral. Specifically, it states: :''Let be a convex quadrilateral with ...

and its converse, any interior point ''P'' on the Newton line of a quadrilateral has the property that :

triangle ABP + triangle CDP =

triangle ADP A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...

+ triangle BCP where denotes the area of triangle . If the quadrilateral is a

tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...

, then its

incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisec ...

also lies on this line.Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović, ''The IMO Compendium'', Springer, 2006, p. 15.

References

External links

*{{MathWorld , urlname=LeonAnnesTheorem , title=Léon Anne's Theorem *

Alexander Bogomolny Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and M ...

''Bimedians in a Quadrilateral''
at cut-the-knot.org Quadrilaterals

Properties

See also

References

External links