# Barrow's inequality

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^{[1]}

- [math]\displaystyle{ PA+PB+PC\geq 2(PU+PV+PW),\, }[/math]

with equality holding only in the case of an equilateral triangle and *P* is the center of the triangle^{[1]}.

## Generalisation

Barrow's inequality can be extended to convex polygons. For a convex polygon with vertices [math]\displaystyle{ A_1,A_2,\ldots ,A_n }[/math] let [math]\displaystyle{ P }[/math] be an inner point and [math]\displaystyle{ Q_1, Q_2,\ldots ,Q_n }[/math] the intersections of the angle bisectors of [math]\displaystyle{ \angle A_1PA_2,\ldots,\angle A_{n-1}PA_n,\angle A_nPA_1 }[/math] with the associated polygon sides [math]\displaystyle{ A_1A_2,\ldots ,A_{n-1}A_n, A_nA_1 }[/math], then the following inequality holds:^{[2]}^{[3]}

- [math]\displaystyle{ \sum_{k=1}^n|PA_k|\geq \sec\left(\frac{\pi}{n}\right) \sum_{k=1}^n|PQ_k| }[/math]

Here [math]\displaystyle{ \sec(x) }[/math] denotes the secant function. For the triangle case [math]\displaystyle{ n=3 }[/math] the inequality becomes Barrow's inequality due to [math]\displaystyle{ \sec\left(\tfrac{\pi}{3}\right)=2 }[/math].

## 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.^{[1]} This result was named "Barrow's inequality" as early as 1961.^{[4]}

A simpler proof was later given by Louis J. Mordell.^{[5]}

## See also

## References

- ↑
^{1.0}^{1.1}^{1.2}"Solution to problem 3740",*American Mathematical Monthly***44**(4): 252–254, 1937, doi:10.2307/2300713. - ↑ 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, doi:10.1007/BF01650566 (German). - ↑ "New inequalities for a triangle and an internal point",
*Annali di Matematica Pura ed Applicata***53**: 157–163, 1961, doi:10.1007/BF02417793 - ↑ "On geometric problems of Erdös and Oppenheim",
*The Mathematical Gazette***46**(357): 213–215, 1962.

## External links