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]

${\displaystyle PA+PB+PC\geq 2(PU+PV+PW),\,}$

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 ${\displaystyle A_{1},A_{2},\ldots ,A_{n}}$ let ${\displaystyle P}$ be an inner point and ${\displaystyle Q_{1},Q_{2},\ldots ,Q_{n}}$ the intersections of the angle bisectors of ${\displaystyle \angle A_{1}PA_{2},\ldots ,\angle A_{n-1}PA_{n},\angle A_{n}PA_{1}}$ with the associated polygon sides ${\displaystyle A_{1}A_{2},\ldots ,A_{n-1}A_{n},A_{n}A_{1}}$, then the following inequality holds: ^{[2] [3]}

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

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

History

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

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

• Euler's theorem in geometry
• List of triangle inequalities

References

1. Erdős, Paul; Mordell, L. J.; Barrow, David F. (1937), "Solution to problem 3740", American Mathematical Monthly , 44 (4): 252–254, doi:10.2307/2300713, JSTOR   2300713 .
2. M. Dinca: "A Simple Proof of the Erdös-Mordell Inequality". In: Articole si Note Matematice, 2009
3. 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).
4. Oppenheim, A. (1961), "New inequalities for a triangle and an internal point", Annali di Matematica Pura ed Applicata , 53: 157–163, doi:10.1007/BF02417793, MR   0124774
5. Mordell, L. J. (1962), "On geometric problems of Erdös and Oppenheim", The Mathematical Gazette , 46 (357): 213–215, JSTOR   3614019 .

External links

• Hojoo Lee: Topics in Inequalities - Theorems and Techniques