# 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.

## 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

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

with equality holding only in the case of an equilateral triangle.

## 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]

A simpler proof was later given by Mordell.[2]

## 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. ^ Mordell, L. J. (1962), "On geometric problems of Erdös and Oppenheim", Mathematical Gazette, 46 (357): 213–215, JSTOR 3614019.