Erdös-Mordell's inequality

Inequality of Erdös and Mordell: }
${\displaystyle d_{A}+d_{B}+d_{C}\geq 2(d_{a}+d_{b}+d_{c})}$

The inequality of Erdos-Mordell , sometimes called set of Erdos-Mordell referred to is an indication of the distances of a point in a triangle of which corners and sides. It says that the sum of the distances from the corners is at least twice the sum of the distances from the sides.

Inequality

The following inequality applies to a point inside a triangle : ${\displaystyle P}$ ${\displaystyle \triangle ABC}$

${\displaystyle |PA|+|PB|+|PC|\geq 2(PF_{a}|+|PF_{b}|+|PF_{c}|)}$.

Here, the base points of the solders from the point to the (extended) triangle sides. ${\displaystyle F_{a},F_{b},F_{c}}$ ${\displaystyle P}$

Equality only occurs if it is an equilateral triangle and is its center of gravity . ${\displaystyle P}$

Generalizations and Related Statements

Erdös-Mordell's inequality can be generalized to convex polygons. For a convex polygon with corner points and a point in its interior designate the base points of the perpendicular to the (extended) sides of the polygon . The following inequality then applies: ${\displaystyle A_{1},A_{2},\ldots ,A_{n}}$ ${\displaystyle P}$ ${\displaystyle F_{1},F_{2},\ldots ,F_{n}}$ ${\displaystyle P}$ ${\displaystyle A_{1}A_{2},\ldots ,A_{n-1}A_{n},A_{n}A_{1}}$

${\displaystyle \sum _{k=1}^{n}|PA_{k}|\geq \sec \left({\frac {\pi }{n}}\right)\sum _{k=1}^{n}|PF_{k}|}$.

Here the secant function is called and in the case one gets exactly the inequality of Erdös-Mordell. ${\displaystyle \sec }$ ${\displaystyle n=3}$ ${\displaystyle \sec \left({\tfrac {\pi }{3}}\right)=2}$

A spatial variant of the Erdös-Mordell inequality can be formulated for the tetrahedron. For a tetrahedron in which the center of its sphere is inside the tetrahedron, denote another inner point of the tetrahedron and the base points of the perpendiculars on the surfaces of the tetrahedron or on the planes in which these surfaces are embedded. The following inequality then applies: ${\displaystyle ABCD}$ ${\displaystyle P}$ ${\displaystyle F_{ABC},F_{ABD},F_{ADC},F_{DBC}}$ ${\displaystyle P}$

${\displaystyle |PA|+|PB|+|PC|+|PD|\geq 2{\sqrt {2}}(|PF_{ABC}|+|PF_{ABD}|+|PF_{ADC}|+|PF_{DBC}|)}$

history

The inequality was described by Paul Erdős as Problem 3740 in the American Mathematical Monthly in 1935 and a first proof was published in a Hungarian journal by Louis Mordell in the same year. David Francis Barrow found a second proof, which also provides a tightening of the inequality ( Barrow's inequality ). Both proofs use trigonometric functions and were published together in the American Mathematical Monthly in 1937. Further simpler elementary geometric proofs come from DK Kazarinoff (1957), Leon Bankoff (1958) and Claudi Alsina / Roger B. Nelsen (2007).

A corresponding inequality for quadrilaterals was proved by A. Florian in 1958. The generalization to convex polygons was initially assumed by László Fejes Tóth in 1961 and then proved by Hans-Christof Lenhard in the same year.

literature

• Wolfgang Zeuge: Useful and beautiful geometry: a slightly different introduction to Euclidean geometry . Springer, 2018, ISBN 9783658228330 , pp. 95-96
• Alexander Ostermann, Gerhard Wanner: Geometry by its History . Springer, 2012, ISBN 9783642291630 , pp. 222-224
• Claudi Alsina, Roger B. Nelsen: A Visual Proof of the Erdos-Mordell Inequality . In: Forum Geometricorum , Volume 7, 2007, pp. 99-102.
• Vilmos Komornik: A Short Proof of the Erdős-Mordell Theorem . In: The American Mathematical Monthly , Volume 104, No. 1 (Jan., 1997), pp. 57-60 ( JSTOR )
• Jian Liu: Refinements of the Erdös-Mordell inequality, Barrow's inequality, and Oppenheim's inequality . In: Journal of Inequalities and Applications , 2016
• Paul Erdös, LJ Mordell, David F. Barrow: 3740 . In: The American Mathematical Monthly , Vol. 44, No. 4 (April, 1937), pp. 252-254 ( JSTOR )

Web links

• Erdös-Mordell Inequality on cut-the-knot.org
• Eric W. Weisstein : Erdos-Mordell Theorem . In: MathWorld (English).

Individual evidence

1. ↑ ^{a b c} Wolfgang Zeuge: Useful and beautiful geometry: A slightly different introduction to Euclidean geometry . Springer, 2018, ISBN 9783658228330 , pp. 95-96
2. ↑ M. Dinca: A Simple Proof of the Erdos-Mordell Inequality . In: Articole si Note Matematice , 2009
3. ↑ ^{a b} Hans-Christof Lenhard: Generalization and tightening of the Erdös-Mordell inequality for polygons . In: Archive for Mathematical Logic and Basic Research , Volume 12, pp. 311-314, doi: 10.1007 / BF01650566, MR 0133060.
4. ↑ ^{a b} Alexander Bogomolny : Erdös-Mordell Inequality on cut-the-knot.org (accessed November 19, 2019)
5. ↑ ^{a b} Vilmos Komornik: A Short Proof of the Erdős-Mordell Theorem . In: The American Mathematical Monthly , Volume 104, No. 1 (Jan., 1997), pp. 57-60 ( JSTOR )