en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

2014-03-29T08:14:12Z

WP:CHECKWIKI error fixes + other fixes using AWB (10065)

81.53.77.230: /* The 11 scoring criteria */

2011-10-06T15:31:25Z

The 11 scoring criteria

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In [[Euclidean geometry]], the '''Erdős–Mordell inequality''' states that for any triangle ''ABC'' and point ''P'' inside ''ABC'', the sum of the distances from ''P'' to the sides is less than or equal to half of the sum of the distances from ''P'' to the vertices. It is named after [[Paul Erdős]] and [[Louis Mordell]]. {{harvtxt|Erdős|1935}} posed the problem of proving the inequality; a proof was provided two years later by {{harvs|last1=Mordell|first2=D. F.|last2=Barrow|year=1937|txt}}. This solution was however not very elementary. Subsequent simpler proofs were then found by {{harvtxt|Kazarinoff|1957}}, {{harvtxt|Bankoff|1958}}, and {{harvtxt|Alsina|Nelson|2007}}.

In [[absolute geometry]], the Erdős–Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is at most 2<math>\pi</math> {{harv|Pambuccian|2008}}.

[[Barrow's inequality]] is a strengthened version of the Erdős–Mordell inequality in which the distances from ''O'' to the sides are replaced by the distances from ''O'' to the points where the [[angle bisector]]s cross the sides. Although the replaced distances are longer, their sum is still less than or equal to half the sum of the distances to the vertices.

== Proof ==

Let the sides of ABC be a, b, c, also let PA=p, PB=q, PC=r, d(P;BC)=x, d(P;CA)=y, d(P;AB)=z. First, we prove that

<math>cr\geq ax+by</math>.

This is equivalent to

<math>\frac{c(r+z)}2\geq \frac{ax+by+cz}2</math>.

The RHS is the area of triangle ABC, but on the LHS, r+z is at least the height of the triangle, consequently, the LHS cannot be smaller than the RHS. Now reflect P on the angle bisector at C. We find that cr<math>\geq</math>ay+bx for P's reflection. Similarly, bq<math>\geq</math>az+cx and ap<math>\geq</math>bz+cy. We solve these inequalities for r, q, and p:

<math>r\geq (a/c)y+(b/c)x</math>,

<math>q\geq (a/b)z+(c/b)x</math>,

<math>p\geq (b/a)z+(c/a)y</math>.

Adding the three up, we get

<math>p+q+r\geq (\frac b c+\frac c b)x+(\frac a c+\frac c a)y+(\frac a b+\frac b a)z</math>.

Since the sum of a positive number and its reciprocal is at least 2, we are finished. Equality holds only for the equilateral triangle, where P is its centroid.

== References ==
*{{citation
| last1 = Alsina | first1 = Claudi
| last2 = Nelsen | first2 = Roger B.
| journal = Forum Geometricorum
| pages = 99–102
| title = A visual proof of the Erdős-Mordell inequality
| url = http://forumgeom.fau.edu/FG2007volume7/FG200711index.html
| volume = 7
| year = 2007}}.
*{{citation
| last = Bankoff | first = Leon | author-link = Leon Bankoff
| journal = [[American Mathematical Monthly]]
| page = 521
| title = An elementary proof of the Erdős-Mordell theorem
| issue = 7
| jstor = 2308580
| volume = 65
| year = 1958}}.
*{{citation
| last = Erdős | first = Paul | author-link = Paul Erdős
| journal = [[American Mathematical Monthly]]
| page = 396
| title = Problem 3740
| volume = 42
| year = 1935}}.
*{{citation
| last = Kazarinoff | first = D. K.
| doi = 10.1307/mmj/1028988998
| issue = 2
| journal = [[Michigan Mathematical Journal]]
| pages = 97–98
| title = A simple proof of the Erdős-Mordell inequality for triangles
| volume = 4
| year = 1957}}.
*{{citation
| last1 = Mordell | first1 = L. J. | author1-link = Louis Mordell
| last2 = Barrow | first2 = D. F.
| journal = [[American Mathematical Monthly]]
| pages = 252–254
| title = Solution to 3740
| volume = 44
| year = 1937}}.
*{{citation
| last = Pambuccian | first = Victor
| doi = 10.1007/s00022-007-1961-4
| journal = Journal of Geometry
| pages = 134–139
| title = The Erdős-Mordell inequality is equivalent to non-positive curvature
| volume = 88
| year = 2008}}.

== External links ==
*{{mathworld|urlname=Erdos-MordellTheorem|title=Erdős-Mordell Theorem}}
*[[Alexander Bogomolny]], "[http://www.cut-the-knot.org/triangle/ErdosMordell.shtml Erdös-Mordell Inequality]", from [[Cut-the-Knot]].

{{DEFAULTSORT:Erdos-Mordell inequality}}
[[Category:Triangle geometry]]
[[Category:Geometric inequalities]]

CCWAPSS - Revision history

en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

81.53.77.230: /* The 11 scoring criteria */