Anne's theorem

Template:Short description In Euclidean geometry, 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).Template:R

Statement

A point Template:Mvar lies on the Newton line if the two sums of the areas of opposing triangles are equal, according to Anne's theorem.

The theorem is stated as follows: Let Template:Mvar be a convex quadrilateral with diagonals Template:Mvar and Template:Mvar, that is not a parallelogram. Furthermore, let Template:Mvar and Template:Mvar be the midpoints of the diagonals, and let Template:Mvar be an arbitrary point in the interior of Template:Mvar, resulting in that Template:Mvar forms four triangles with the edges of Template:Mvar. If the two sums of areas of opposite triangles are equal: $$\left|\mathrm{△}BCL\right|+\left|\mathrm{△}DAL\right|=\left|\mathrm{△}LAB\right|+\left|\mathrm{△}DLC\right|,$$ then the point Template:Mvar is located on the Newton line, that is the line which connects Template:Mvar and Template:Mvar.Template:R

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.

External links

• Newton's and Léon Anne's Theorems at cut-the-knot.org
• Andrew Jobbings: The Converse of Leon Anne's Theorem Template:Webarchive
• Template:MathWorld