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 $\angle A_{1}PA_{2},\ldots ,\angle A_{n-1}PA_{n},\angle A_{n}PA_{1}$ with the associated polygon sides $A_{1}A_{2},\ldots ,A_{n-1}A_{n},A_{n}A_{1}$, then the following inequality holds:

$\sum _{k=1}^{n}|PA_{k}|\geq \sec \left({\frac {\pi }{n}}\right)\sum _{k=1}^{n}|PQ_{k}|$

Here $\sec(x)$ denotes the secant function. For the triangle case $n=3$ the inequality becomes Barrow's inequality due to $\sec \left({\tfrac {\pi }{3}}\right)=2$.

History

Barrow strengthening Erdős-Mordell
${\begin{aligned}&\quad \,|PA|+|PB|+|PC|\\&\geq 2(|PQ_{a}|+|PQ_{b}|+|PQ_{c}|)\\&\geq 2(|PF_{a}|+|PF_{b}|+|PF_{c}|)\end{aligned}}$

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

External links

• Hojoo Lee: Topics in Inequalities - Theorems and Techniques