Barrow's inequality

Barrow as a tightening of Erdös-Mordell

{\ displaystyle {\ begin {aligned} & \ quad \, | PA | + | PB | + | PC | \\ & \ geq 2 (| PQ_ {a} | + | PQ_ {b} | + | PQ_ {c } |) \\ & \ geq 2 (| PF_ {a} | + | PF_ {b} | + | PF_ {c} |) \ end {aligned}}}

The inequality of Barrow , named after David Francis Barrow is a statement about distances of a point inside a triangle at specific points on its edge. It represents an exacerbation of Erdös-Mordell's inequality , which states that the sum of the distances from the corners is always at least twice as large as the sum of the distances from the sides.

Inequality

Any point inside a triangle is given . The bisectors of the angles , and intersect the sides of the triangle in the points . The following inequality then applies: ${\ displaystyle P}$ ${\ displaystyle \ triangle ABC}$ ${\ displaystyle \ angle APB}$ ${\ displaystyle \ angle BPC}$ ${\ displaystyle \ angle CPA}$ ${\ displaystyle a, b, c}$ ${\ displaystyle Q_ {a}, Q_ {b}, Q_ {c}}$

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

.

In 1937 Barrow published a proof of Erdös-Mordell's inequality, which contained the inequality later named after him as an intermediate step.

generalization

Barrow's inequality can be generalized to convex polygons. For a convex polygon with corner points and a point in its interior designate the points that arise when intersecting with the bisectors of the angles with polygon sides . The following inequality then applies: ${\ displaystyle A_ {1}, A_ {2}, \ ldots, A_ {n}}$ ${\ displaystyle P}$ ${\ displaystyle Q_ {1}, Q_ {2}, \ ldots, Q_ {n}}$ ${\ displaystyle \ angle A_ {1} PA_ {2}, \ ldots, \ angle A_ {n-1} PA_ {n}, \ angle A_ {n} PA_ {1}}$ ${\ displaystyle A_ {1} A_ {2}, \ ldots, A_ {n-1} A_ {n}, A_ {n} A_ {1}}$

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

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Here denotes the secant function and in this case one obtains exactly the inequality of Barrow because of. ${\ displaystyle \ sec}$ ${\ displaystyle n = 3}$ ${\ displaystyle \ sec \ left ({\ tfrac {\ pi} {3}} \ right) = 2}$

literature

Jian Liu: Sharpened versions of the Erdös-Mordell inequality . In: Journal of Inequalities and Applications , Issue 1, 2015
Alexander Ostermann, Gerhard Wanner: Geometry by its History . Springer, 2012, ISBN 978-3-642-29163-0 , pp. 222-224
Branko Malesevic, Maja Petrovic: Barrow's Inequality and Signed Angle Bisectors . In: Journal of Mathematical Inequalities , Volume 8, No. 3, 2014
Paul Erdös, LJ Mordell, David F. Barrow: Solution to 3740 . In: The American Mathematical Monthly , Vol. 44, No. 4 (April, 1937), pp. 252-254 ( JSTOR )

Individual evidence

^ ^A ^b Alexander Ostermann, Gerhard Wanner: Geometry by its History . Springer, 2012, ISBN 978-3-642-29163-0 , pp. 222-224
↑ M. Dinca: A Simple Proof of the Erdos-Mordell Inequality . In: Articole si Note Matematice , 2009
↑ 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.

[OW-1] A ^b Alexander Ostermann, Gerhard Wanner: Geometry by its History . Springer, 2012, ISBN 978-3-642-29163-0 , pp. 222-224

[2] M. Dinca: A Simple Proof of the Erdos-Mordell Inequality . In: Articole si Note Matematice , 2009

[3] 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.