háromszöginegualitást

The triangle inequality is a fundamental concept in geometry and mathematics. It states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. If we denote the lengths of the sides of a triangle as a, b, and c, then the triangle inequality can be expressed as three inequalities: a + b > c, a + c > b, and b + c > a.

This principle holds true for all types of triangles, whether they are Euclidean, spherical, or hyperbolic. In Euclidean geometry, the triangle inequality is directly derived from the properties of straight lines. If the sum of two sides were equal to or less than the third side, the three points would not form a triangle but would instead lie on a straight line or not form a closed figure at all.

The triangle inequality has broad applications beyond simple triangles. It is a key property in defining distance metrics in various mathematical spaces. For instance, in linear algebra, the norm of a vector satisfies the triangle inequality, meaning the "distance" between two vectors adheres to this rule. Similarly, in functional analysis and topology, metrics used to measure distances between functions or points in abstract spaces must satisfy this inequality. It is a foundational axiom for any function to be considered a valid distance metric.