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In
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, 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 Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, g ...
and Louis Mordell. posed the problem of proving the inequality; a proof was provided two years later by . This solution was however not very elementary. Subsequent simpler proofs were then found by , , and . Barrow's inequality is a strengthened version of the Erdős–Mordell inequality in which the distances from ''P'' to the sides are replaced by the distances from ''P'' to the points where the
angle bisector In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''se ...
s of ∠''APB'', ∠''BPC'', and ∠''CPA'' 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.


Statement

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\geq 2(PL+PM+PN)


Proof

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\geq ax+by. This is equivalent to :\frac2\geq \frac2. 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\geq (a/c)y+(b/c)x, :q\geq (a/b)z+(c/b)x, :p\geq (b/a)z+(c/a)y. Adding the three up, we get : p + q + r \geq \left( \frac + \frac \right) x + \left( \frac + \frac \right) y + \left( \frac + \frac \right) z. Since the sum of a positive number and its reciprocal is at least 2 by
AM–GM inequality In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...
, we are finished. Equality holds only for the equilateral triangle, where P is its centroid.


Another strengthened version

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 \ge 2(PD+PE+PF) Equality hold if and only if triangle ABC is equilateral (; )


A generalization

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_, w_i the segment of the bisector of the angle A_iPA_ from P to its intersection with the side A_iA_ then : : \sum_^R_i \ge \left(\sec\right)\sum_^ w_i \ge \left(\sec\right)\sum_^ r_i


In absolute geometry

In
absolute geometry Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates. The term was introduced by ...
the Erdős–Mordell inequality is equivalent, as proved in , to the statement that the sum of the angles of a triangle is less than or equal to two right angles.


See also

* List of triangle inequalities


References

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


External links

* * Alexander Bogomolny,
Erdös-Mordell Inequality
, from
Cut-the-Knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...
. {{DEFAULTSORT:Erdos-Mordell inequality Triangle inequalities