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geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, Barrow's inequality is an inequality relating the distances between an arbitrary point within a
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, 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 A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
''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 : PA+PB+PC\geq 2(PU+PV+PW),\, with equality holding only in the case of an
equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
and ''P'' is the center of the triangle.


Generalisation

Barrow's inequality can be extended to convex polygons. For a convex polygon with vertices A_1,A_2,\ldots ,A_n let P be an inner point and Q_1, Q_2,\ldots ,Q_n the intersections of the angle bisectors of \angle A_1PA_2,\ldots,\angle A_PA_n,\angle A_nPA_1 with the associated polygon sides A_1A_2,\ldots ,A_A_n, A_nA_1, then the following inequality holds: :\sum_^n, PA_k, \geq \sec\left(\frac\right) \sum_^n, PQ_k, Here \sec(x) denotes the secant function. For the triangle case n=3 the inequality becomes Barrow's inequality due to \sec\left(\tfrac\right)=2.


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 ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...
of proving the Erdős–Mordell inequality.. This result was named "Barrow's inequality" as early as 1961. A simpler proof was later given by Louis J. Mordell..


See also

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


References

{{reflist


External links


Hojoo Lee: Topics in Inequalities - Theorems and Techniques
Triangle inequalities