# Barrow's inequality

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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 $\displaystyle{ \angle A_1PA_2,\ldots,\angle A_{n-1}PA_n,\angle A_nPA_1 }$ with the associated polygon sides $\displaystyle{ A_1A_2,\ldots ,A_{n-1}A_n, A_nA_1 }$, 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 Erdös-Mordell
\displaystyle{ \begin{align}&\quad\, |PA|+|PB|+|PC| \\ &\ge 2 (|PQ_a|+|PQ_b|+|PQ_c|)\\ &\ge 2 (|PF_a|+|PF_b|+|PF_c|)\end{align} }

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]