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 ...

, Anne's theorem describes an equality of certain areas within a convex quadrilateral. This theorem is named after the French mathematician Pierre-Léon Anne (1806–1850).

Statement

The theorem is stated as follows: Let be a convex quadrilateral with diagonals and , that is not a

parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

. Furthermore, let and 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''-dim ...

s of the diagonals, and let be an arbitrary point in the interior of , resulting in that forms four

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

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

\left, \triangle BCL \ + \left, \triangle DAL \ = \left, \triangle LAB \ + \left, \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. Properties The line segments and that connect the midpoints of opposite sides (the ...

, 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

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