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The reverse triangle inequality is an equivalent alternative formulation of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is: [ 19 ] Any side of a triangle is greater than or equal to the difference between the other two sides .
The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);
Pages in category "Triangle inequalities" The following 8 pages are in this category, out of 8 total. This list may not reflect recent changes. *
Rewriting the inequality above allows for a more concrete geometric interpretation, which in turn provides an immediate proof. [1]+ +. Now the summands on the left side are the areas of equilateral triangles erected over the sides of the original triangle and hence the inequation states that the sum of areas of the equilateral triangles is always greater than or equal to threefold the area of ...
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.)
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.
The Huffington Post collaborated with The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education, to produce this story, the first in a series on the impact of police in schools. Kyle Spencer is a reporter for The Hechinger Report and Adam Hooper is a data editor for The Huffington Post.
This inequality fails for general triangles (to which Ono's original conjecture applied), as shown by the counterexample =, =, =, = / The inequality holds with equality in the case of an equilateral triangle , in which up to similarity we have sides 1 , 1 , 1 {\displaystyle 1,1,1} and area 3 / 4. {\displaystyle {\sqrt {3}}/4.}
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