Anne's theorem

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).^[1]

Statement

The theorem is stated as follows: Let $ABCD$ be a convex quadrilateral with diagonals $AC$ and $BD$ , that is not a parallelogram. Furthermore, let $E$ and $F$ be the midpoints of the diagonals, and let $L$ be an arbitrary point in the interior of $ABCD$ , resulting in that $L$ forms four triangles with the edges of $ABCD$ . If the two sums of areas of opposite triangles are equal: $\left|\triangle BCL\right|+\left|\triangle DAL\right|=\left|\triangle LAB\right|+\left|\triangle DLC\right|,$ then the point $L$ is located on the Newton line, that is the line which connects $E$ and $F$ .^[1]^[2]

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

^ ^a ^b Alsina, Claudi; Nelsen, Roger B. (2020). A Cornucopia of Quadrilaterals. American Mathematics Society. pp. 12–13. ISBN 9781470454654.
^ Honsberger, Ross (1991). More Mathematical Morsels. Cambridge University Press. pp. 174–175. ISBN 0883853140.

External links

Newton's and Léon Anne's Theorems at cut-the-knot.org
Andrew Jobbings: The Converse of Leon Anne's Theorem Archived 2014-03-04 at the Wayback Machine
Weisstein, Eric W. "Leon Anne's Theorem". MathWorld.

[Alsina-1] Alsina, Claudi; Nelsen, Roger B. (2020). A Cornucopia of Quadrilaterals. American Mathematics Society. pp. 12–13. ISBN 9781470454654.

[Honsberger-2] Honsberger, Ross (1991). More Mathematical Morsels. Cambridge University Press. pp. 174–175. ISBN 0883853140.

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