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In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proved without these theorems.
In geometry, triangle inequalities ... The parameters in a triangle inequality can ... The in-between case of equality when C is a right angle is the Pythagorean theorem.
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 :
In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. [1 ...
Kunita–Watanabe inequality; Le Cam's theorem; Lenglart's inequality; Marcinkiewicz–Zygmund inequality; Markov's inequality; McDiarmid's inequality; Paley–Zygmund inequality; Pinsker's inequality; Popoviciu's inequality on variances; Prophet inequality; Rao–Blackwell theorem; Ross's conjecture, a lower bound on the average waiting time ...
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. [48] Conversely, some triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. [49]
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.)
Pages in category "Triangle inequalities" The following 8 pages are in this category, out of 8 total. ... Erdős–Mordell inequality; Euler's theorem in geometry; H.
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