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: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, 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 ( hu, Erdős Pál ; 26 March 1913 – 20 September 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 ...
and
Louis Mordell
Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction.
Educati ...
. 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
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.
Stateme ...
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, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...
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
be an arbitrary point P inside a given triangle
, and let
,
, and
be the perpendiculars from
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
:
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
:
This is equivalent to
:
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'':
:
:
:
Adding the three up, we get
:
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; a ...
, 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:
:
Equality hold if and only if triangle ABC is equilateral (; )
A generalization
Let
be a convex polygon, and
be an interior point of
. Let
be the distance from
to the vertex
,
the distance from
to the side
,
the segment of the bisector of the angle
from
to its intersection with the side
then :
:
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, but since these are not su ...
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
In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of th ...
References
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External links
*
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Alexander Bogomolny
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and M ...
,
Erdös-Mordell Inequality, from
Cut-the-Knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
.
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Triangle inequalities