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<div>[[Image:Barrow inequality.svg|thumb|right|300px]]<br />
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In [[geometry]], '''Barrow's inequality''' is an [[Inequality (mathematics)|inequality]] relating the [[Euclidean distance|distances]] between an arbitrary point within a [[triangle]], the vertices of the triangle, and certain points on the sides of the triangle.<br />
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==Statement==<br />
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 bisector]]s of ''BPC'', ''CPA'', and ''APB'' intersect the sides ''BC'', ''CA'', ''AB'', respectively. Then Barrow's inequality states that<br />
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: <math>PA+PB+PC\geq 2(PU+PV+PW),\,</math><br />
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with equality holding only in the case of an [[equilateral triangle]].<br />
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==History==<br />
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.<ref>{{citation<br />
| last1 = Erdős | first1 = Paul | author1-link = Paul Erdős<br />
| last2 = Mordell | first2 = L. J. | author2-link = Louis J. Mordell<br />
| last3 = Barrow | first3 = David F. | author3-link = David Francis Barrow<br />
| issue = 4<br />
| journal = [[American Mathematical Monthly]]<br />
| jstor = 2300713<br />
| pages = 252–254<br />
| title = Solution to problem 3740<br />
| volume = 44<br />
| year = 1937}}.</ref> A simpler proof was later given by Mordell.<ref>{{citation<br />
| last = Mordell | first = L. J. | author-link = Louis J. Mordell<br />
| issue = 357<br />
| journal = Mathematical Gazette<br />
| jstor = 3614019<br />
| pages = 213–215<br />
| title = On geometric problems of Erdös and Oppenheim<br />
| volume = 46<br />
| year = 1962}}.</ref><br />
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== See also ==<br />
* [[Euler's theorem in geometry]]<br />
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==References==<br />
{{reflist}}<br />
<br />
==External links==<br />
*[http://www.eleves.ens.fr/home/kortchem/olympiades/Cours/Inegalites/tin2006.pdf Hojoo Lee: Topics in Inequalities - Theorems and Techniques]<br />
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[[Category:Triangle geometry]]<br />
[[Category:Geometric inequalities]]</div>96.11.61.34