Theorem of Anne
The set of Anne , named after Pierre-Leon Anne (1806-1850) is a statement of the elementary geometry , a certain decomposition of a convex quadrilateral describes in equal areas.
The following statement applies more precisely:
- Let ABCD be a convex rectangle with the diagonals AC and BD that is not a parallelogram. Furthermore, let E and F be the midpoints of the diagonals and L a point inside ABCD. For the four triangles that the point L forms with the sides of ABCD, the two sums of the areas of opposing triangles are equal (F (BCL) + F (DAL) = F (LAB) + F (DLC)), so the point L lies on the Newton straight line , that is, the straight line which connects the centers of the diagonals AC and BD.
In the case of a parallelogram, there is no Newton straight line because the diagonal centers coincide to one point. In addition, in this case the condition of the equality of the area sums of opposing triangles is fulfilled by every inner point.
The reverse of Anne's theorem also applies, that is, for every point on the Newton line that lies within the associated quadrilateral, the condition of equality of area is fulfilled.
literature
- Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics . MAA, 2010, ISBN 9780883853481 , pp. 116–117 ( excerpt (Google) )
- Ross Honsberger: More Mathematical Morsels . Cambridge University Press, 1991, ISBN 0883853140 , pp. 174–175 ( excerpt (Google) )
Web links
- Newton's and Léon Anne's theorems on cut-the-knot.org
- Andrew Jobbings: The Converse of Leon Anne's Theorem
- Leon Anne's Theorem on MathWorld