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