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

, 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

diagonals 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 ...

and , that is not a parallelogram. Furthermore let be the

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimen ...

s of the diagonals and be an arbitrary point in the interior of . forms four

triangle 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- colline ...

s with the edges of . If the two sums of areas of opposite triangles are equal:'' :::

, \triangle BCL, \ +\ , \triangle DAL, \ =\ , \triangle LAB, \ +\ , \triangle DLC,

:''then the point is located on the

Newton line In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides.Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. ...

, that is the line which connects and .'' For a parallelogram the Newton line does not exist since both midpoints of the diagonals coincide with point of intersection of the diagonals. Moreover the area identity of the theorem holds in this case for any inner point of the quadrilateral. The converse of Anne's theorem is true as well, that is for any point on the Newton line which is an inner point of the quadrilateral, the area identity holds.

References

*Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. MAA, 2010, , pp. 116–117 () *Ross Honsberger: ''More Mathematical Morsels''. Cambridge University Press, 1991, , pp. 174–175 )

External links

''Newton's and Léon Anne's Theorems''
at cut-the-knot.org *Andrew Jobbings
''The Converse of Leon Anne's Theorem''
*{{MathWorld, urlname=LeonAnnesTheorem, title=Leon Anne's Theorem Theorems about quadrilaterals