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[1]

P A + P B + P C ≥ 2 ( P U + P V + P W ) , {\displaystyle 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.[1]

Generalisation

Barrow's inequality can be extended to convex polygons. For a convex polygon with vertices A 1 , A 2 , … , A n {\displaystyle A_{1},A_{2},\ldots ,A_{n}} let P {\displaystyle P} be an inner point and Q 1 , Q 2 , … , Q n {\displaystyle Q_{1},Q_{2},\ldots ,Q_{n}} the intersections of the angle bisectors of ∠ A 1 P A 2 , … , ∠ A n − 1 P A n , ∠ A n P A 1 {\displaystyle \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 , … , A n − 1 A n , A n A 1 {\displaystyle A_{1}A_{2},\ldots ,A_{n-1}A_{n},A_{n}A_{1}} , then the following inequality holds:[2][3]

∑ k = 1 n | P A k | ≥ sec ⁡ ( π n ) ∑ k = 1 n | P Q k | {\displaystyle \sum _{k=1}^{n}|PA_{k}|\geq \sec \left({\frac {\pi }{n}}\right)\sum _{k=1}^{n}|PQ_{k}|}

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

History

Barrow strengthening Erdös-Mordell
| P A | + | P B | + | P C | ≥ 2 ( | P Q a | + | P Q b | + | P Q c | ) ≥ 2 ( | P F a | + | P F b | + | P F c | ) {\displaystyle {\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.[1] This result was named "Barrow's inequality" as early as 1961.[4]

A simpler proof was later given by Louis J. Mordell.[5]

See also

References

  1. ^ a b c Erdős, Paul; Mordell, L. J.; Barrow, David F. (1937), "Solution to problem 3740", American Mathematical Monthly, 44 (4): 252–254, doi:10.2307/2300713, JSTOR 2300713.
  2. ^ M. Dinca: "A Simple Proof of the Erdös-Mordell Inequality". In: Articole si Note Matematice, 2009
  3. ^ Hans-Christof Lenhard: "Verallgemeinerung und Verschärfung der Erdös-Mordellschen Ungleichung für Polygone". In: Archiv für Mathematische Logik und Grundlagenforschung, Band 12, S. 311–314, doi:10.1007/BF01650566 (German).
  4. ^ Oppenheim, A. (1961), "New inequalities for a triangle and an internal point", Annali di Matematica Pura ed Applicata, 53: 157–163, doi:10.1007/BF02417793, MR 0124774
  5. ^ Mordell, L. J. (1962), "On geometric problems of Erdös and Oppenheim", The Mathematical Gazette, 46 (357): 213–215, JSTOR 3614019.

External links

  • Hojoo Lee: Topics in Inequalities - Theorems and Techniques