Barrow's Inequality

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

: $PA+PB+PC\geq 2(PU+PV+PW),\,$

with equality holding only in the case of an equilateral triangle and ''P'' is the center of the triangle.

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 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