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]
- [math]\displaystyle{ PA+PB+PC\geq 2(PU+PV+PW),\, }[/math]
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 [math]\displaystyle{ A_1,A_2,\ldots ,A_n }[/math] let [math]\displaystyle{ P }[/math] be an inner point and [math]\displaystyle{ Q_1, Q_2,\ldots ,Q_n }[/math] the intersections of the angle bisectors of [math]\displaystyle{ \angle A_1PA_2,\ldots,\angle A_{n-1}PA_n,\angle A_nPA_1 }[/math] with the associated polygon sides [math]\displaystyle{ A_1A_2,\ldots ,A_{n-1}A_n, A_nA_1 }[/math], then the following inequality holds:[2][3]
- [math]\displaystyle{ \sum_{k=1}^n|PA_k|\geq \sec\left(\frac{\pi}{n}\right) \sum_{k=1}^n|PQ_k| }[/math]
Here [math]\displaystyle{ \sec(x) }[/math] denotes the secant function. For the triangle case [math]\displaystyle{ n=3 }[/math] the inequality becomes Barrow's inequality due to [math]\displaystyle{ \sec\left(\tfrac{\pi}{3}\right)=2 }[/math].
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.[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
- ↑ 1.0 1.1 1.2 "Solution to problem 3740", American Mathematical Monthly 44 (4): 252–254, 1937, doi:10.2307/2300713.
- ↑ M. Dinca: "A Simple Proof of the Erdös-Mordell Inequality". In: Articole si Note Matematice, 2009
- ↑ 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).
- ↑ "New inequalities for a triangle and an internal point", Annali di Matematica Pura ed Applicata 53: 157–163, 1961, doi:10.1007/BF02417793
- ↑ "On geometric problems of Erdös and Oppenheim", The Mathematical Gazette 46 (357): 213–215, 1962.
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