Barrow's inequality explained

In geometry, Barrow's inequality is an inequality relating the distances between an arbitrary point within a triangle, the vertices of the triangle, and certain points on the sides of the triangle. It is named after David Francis Barrow.

Statement

Let P be an arbitrary point inside the triangle ABC. From P and ABC, define U, V, and W as the points where the angle bisectors of BPC, CPA, and APB intersect the sides BC, CA, AB, respectively. Then Barrow's inequality states that

PA+PB+PC\geq2(PU+PV+PW),

with equality holding only in the case of an equilateral triangle and P is the center of the triangle.

Generalisation

Barrow's inequality can be extended to convex polygons. For a convex polygon with vertices

A_1,A_2,\ldots,A_n

let

be an inner point and

Q_1,Q_2,\ldots,Q_n

the intersections of the angle bisectors of

\angleA_1PA_{2,\ldots,\angle}A_n-1PA_n,\angleA_nPA₁

with the associated polygon sides

A_1A_2,\ldots,A_n-1A_n,A_nA₁

, then the following inequality holds:^[1] ^[2]

	n\|PA
\sum		\sec\left(
	k\|\geq

	\pi
	n

\right)

	n\|PQ
\sum
	k\|

Here

\sec(x)

denotes the secant function. For the triangle case

n=3

the inequality becomes Barrow's inequality due to

\sec\left(\tfrac{\pi}{3}\right)=2

History

Barrow's inequality strengthens the Erdős–Mordell inequality, which has identical form except with PU, PV, and PW replaced by the three distances of P from the triangle's sides. It is named after David Francis Barrow. Barrow's proof of this inequality was published in 1937, as his solution to a problem posed in the American Mathematical Monthly of proving the Erdős–Mordell inequality.^[3] This result was named "Barrow's inequality" as early as 1961.

A simpler proof was later given by Louis J. Mordell.^[4]

External links

Hojoo Lee: Topics in Inequalities - Theorems and Techniques

Notes and References

M. Dinca: "A Simple Proof of the Erdös-Mordell Inequality". In: Articole si Note Matematice, 2009
Hans-Christof Lenhard: "Verallgemeinerung und Verschärfung der Erdös-Mordellschen Ungleichung für Polygone". In: Archiv für Mathematische Logik und Grundlagenforschung, Band 12, S. 311–314, (German).
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