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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 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);
In this example, the triangle's side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well when the side lengths are real numbers. As long as they obey the strict triangle inequality, they define a triangle in the Euclidean plane whose area is a positive real number.
Hadwiger–Finsler inequality; Hinge theorem; Hitchin–Thorpe inequality; Isoperimetric inequality; Jordan's inequality; Jung's theorem; Loewner's torus inequality; Łojasiewicz inequality; Loomis–Whitney inequality; Melchior's inequality; Milman's reverse Brunn–Minkowski inequality; Milnor–Wood inequality; Minkowski's first inequality ...
Triangle inequalities (8 P) Pages in category "Theorems about triangles" The following 29 pages are in this category, out of 29 total. ... Marden's theorem; Maxwell's ...
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.
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 ...
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 .