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The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area ...
The basic triangle inequality is +, +, + or equivalently (,,).. In addition, + + + + + <, where the value of the right side is the lowest possible bound, [1]: p. 259 approached asymptotically as certain classes of triangles approach the degenerate case of zero area.
The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range. Using the Reverse Minkowski, we may prove that power means with p ≤ 1 , {\textstyle p\leq 1,} such as the harmonic mean and the geometric mean are concave.
Triangle inequality: If a, b, and c are the lengths of the sides of a triangle then the triangle inequality states that +, with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths :
Erdős–Mordell inequality. 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.)
For four points in order around a circle, Ptolemy's inequality becomes an equality, known as Ptolemy's theorem: ¯ ¯ + ¯ ¯ = ¯ ¯. In the inversion-based proof of Ptolemy's inequality, transforming four co-circular points by an inversion centered at one of them causes the other three to become collinear, so the triangle equality for these three points (from which Ptolemy's inequality may ...
Hadwiger–Finsler inequality is actually equivalent to Weitzenböck's inequality. Applying (W) to the circummidarc triangle gives (HF) [1] Weitzenböck's inequality can also be proved using Heron's formula, by which route it can be seen that equality holds in (W) if and only if the triangle is an equilateral triangle, i.e. a = b = c.
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]
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