Erdős–Mordell inequality

File:Ungleichung erdos mordell2.svg

Let $P$ be an arbitrary point P inside a given triangle $ABC$, and let $PL$, $PM$, and $PN$ be the perpendiculars from $P$ to the sides of the triangles.

(If the triangle is obtuse, one of these perpendiculars may cross through a different side of the triangle and end on the line supporting one of the sides.) Then the inequality states that

:$PA+PB+PC\backslash geq\; 2(PL+PM+PN)$

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

:$cr\backslash geq\; ax+by.$

This is equivalent to

:$\backslash frac\{c(r+z)\}2\backslash geq\; \backslash frac\{ax+by+cz\}2.$

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

:$r\backslash geq\; (a/c)y+(b/c)x,$

:$q\backslash geq\; (a/b)z+(c/b)x,$

:$p\backslash geq\; (b/a)z+(c/a)y.$

Adding the three up, we get

:$$

p + q + r

\geq

\left( \frac{b}{c} + \frac{c}{b} \right) x +

\left( \frac{a}{c} + \frac{c}{a} \right) y +

\left( \frac{a}{b} + \frac{b}{a} \right) z.

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

Let ABC be a triangle inscribed into a circle (O) and P be a point inside of ABC. Let D, E, F be the orthogonal projections of P onto BC, CA, AB. M, N, Q be the orthogonal projections of P onto tangents to (O) at A, B, C respectively, then:

:$PM+PN+PQ\; \backslash ge\; 2(PD+PE+PF)$

Equality hold if and only if triangle ABC is equilateral ({{harvnb|Dao|Nguyen|Pham|2016}}; {{harvnb|Marinescu|Monea|2017}})

Let $A\_1A\_2...A\_n$ be a convex polygon, and $P$ be an interior point of $A\_1A\_2...A\_n$. Let $R\_i$ be the distance from $P$ to the vertex $A\_i$ , $r\_i$ the distance from $P$ to the side $A\_iA\_\{i+1\}$, $w\_i$ the segment of the bisector of the angle $A\_iPA\_\{i+1\}$ from $P$ to its intersection with the side $A\_iA\_\{i+1\}$ then {{harv|Lenhard|1961}}:

:$\backslash sum\_\{i=1\}^\{n\}R\_i\; \backslash ge\; \backslash left(\backslash sec\{\backslash frac\{\backslash pi\}\{n\}\}\backslash right)\backslash sum\_\{i=1\}^\{n\}\; w\_i\; \backslash ge\; \backslash left(\backslash sec\{\backslash frac\{\backslash pi\}\{n\}\}\backslash right)\backslash sum\_\{i=1\}^\{n\}\; r\_i$

In absolute geometry the Erdős–Mordell inequality is equivalent, as proved in {{harvtxt|Pambuccian|2008}}, to the statement

that the sum of the angles of a triangle is less than or equal to two right angles.

• List of triangle inequalities

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| url = http://forumgeom.fau.edu/FG2007volume7/FG200711index.html

| volume = 7

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• {{citation

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• {{citation

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| mr = 3556993

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| title = A strengthened version of the Erdős-Mordell inequality

| volume = 16

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| url = http://forumgeom.fau.edu/FG2016volume16/FG201638.pdf}}.

• {{citation

| last = Erdős | first = Paul | author-link = Paul Erdős

| journal = American Mathematical Monthly

| page = 396

| title = Problem 3740

| volume = 42

| year = 1935

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• {{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| doi-access = free

}}. (See D. K. Kazarinoff's inequality for tetrahedra.)

• {{citation

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| journal = Archiv für Mathematische Logik und Grundlagenforschung

| mr = 0133060

| pages = 311–314

| title = Verallgemeinerung und Verschärfung der Erdös-Mordellschen Ungleichung für Polygone

| volume = 12

| year = 1961| s2cid = 124681241

}}.

• {{citation

| last1 = Marinescu | first1 = Dan Ștefan

| last2 = Monea | first2 = Mihai

| title = About a strengthened version of the Erdős-Mordell inequality

| journal = Forum Geometricorum

| volume = 17

| year = 2017

| pages = 197–202

| url = http://forumgeom.fau.edu/FG2017volume17/FG201723.pdf}}.

• {{citation

| last1 = Mordell | first1 = L. J. | author1-link = Louis Mordell

|author-link2=David Francis Barrow| last2 = Barrow | first2 = D. F.

| journal = American Mathematical Monthly

| pages = 252–254

| title = Solution to 3740

| volume = 44

| year = 1937 | doi=10.2307/2300713| jstor = 2300713 }}.

• {{citation

| last1 = Pambuccian | first1 = Victor

| journal = Journal of Geometry

| pages = 134–139

| title = The Erdős-Mordell inequality is equivalent to non-positive curvature

| volume = 88

| year = 2008 | issue = 1–2

| doi=10.1007/s00022-007-1961-4| s2cid = 123082256

}}.

• {{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 inequalities