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

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

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 $A_1,A_2,\dots ,A_n$ let $P$ be an inner point and $Q_1,Q_2,\dots ,Q_n$ the intersections of the angle bisectors of $\mathrm{\angle }{A}_{1}P{A}_2,\dots ,\mathrm{\angle }{A}_{n-1}P{A}_n,\mathrm{\angle }{A}_{n}P{A}_1$ with the associated polygon sides $A_1{A}_2,\dots ,A_{n-1}{A}_n,A_n{A}_1$, then the following inequality holds:

$\sum _{k=1}^n|P{A}_k|\ge \sec \left(\frac{\pi }{n}\right)\sum _{k=1}^n|P{Q}_k|$

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

History

Barrow strengthening Erdős-Mordell
$\begin{array}{cc}& |PA|+|PB|+|PC|\\ & \ge 2(|P{Q}_a|+|P{Q}_b|+|P{Q}_c|)\\ & \ge 2(|P{F}_a|+|P{F}_b|+|P{F}_c|)\end{array}$

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