Reverse Triangle Inequality

In mathematics, an inequality known as the triangle inequality asserts that the total of the lengths of any two sides of a triangle must be more than or equal to the length of the third side of the triangle. This rule applies to any triangle. This statement allows for the inclusion of degenerate triangles; however, some authors, particularly those writing about elementary geometry, will exclude this possibility, leaving the possibility of equality out of the picture. This statement allows for the inclusion of degenerate triangles; however, some authors, particularly those writing about elementary geometry. The triangle inequality asserts that if the lengths of the sides of the triangle are x, y, and z, respectively, and no side is longer than z, then the triangle inequality is satisfied.

Z ≤ x + y

Only in the degenerate situation of a triangle with 0° area does equality hold, but only then. The triangle inequality is a theorem concerning distances that is used in Euclidean geometry and in certain other geometries. It is expressed using vectors and vector lengths (norms), and the formula is as follows:

|| x + y || ≤ || x || + || y ||

Where the sum of the vectors x + y has been substituted for the length z that was originally on the third side. If x + y are real numbers, then they may be represented as vectors in R1, and the triangle inequality can be seen as expressing a connection between two different absolute values.

In Euclidean geometry, the triangle inequality is a consequence of the Pythagorean Theorem for right triangles and a consequence of the law of cosines for general triangles, despite the fact that it can be shown without these theorems. This is true for both right triangles and generic triangles. Either the R² or R³ representation of the inequality may be intuitively understood. The illustration on the right depicts three scenarios, each of which begins with a glaring disparity and moves closer and closer to parity (bottom). In the Euclidean system, equality can only be achieved if the triangle in question has an angle of 180° as well as two angles of 0°. This configuration will result in the triangle’s three vertices being collinear, as seen in the bottom illustration. As a result, in Euclidean geometry, a straight line represents the shortest distance that can be travelled between two locations.

Reverse Triangle Inequality

An essential consequence of the triangle inequality, the reverse triangle inequality provides lower boundaries rather than higher bounds when applied to a triangle inequality. Any one side of a triangle has a length that is more than or equal to the distance that separates the other two sides.

One of the fundamental concepts in mathematics that is applicable to a wide range of subfields is known as the triangle inequality theorem. Due to the fact that their line of work involves surveying, transportation, and urban planning, civil engineers often use the triangle inequality theorem in the actual world. They are able to determine the unknown lengths with the assistance of the triangle inequality theorem, which also provides them with a rough approximation of the various dimensions. In this article, we will study the triangle inequality theorem and its proof by looking at instances of problems that have already been addressed.

Proof of Reverse triangle Inequality theorem

The relationship that exists between a triangle’s three sides is analysed by the theorem known as the triangle inequality. This theorem states that the total of the lengths of any two of a triangle’s sides is always going to be longer than the third side, regardless of the shape of the triangle. In other words, according to this theorem, a straight line is always going to be the shortest distance that can be travelled between two different sites.

Let’s assume, without sacrificing generality, that ||x|| does not measure up to ||y|| in terms of size. (In the event that this does not occur, we simply switch the roles of x and y.) Therefore, we need to demonstrate that.

|| x || – || y || ≤ || x – y ||

This comes directly from the triangle inequality if we write x as,

x=x-y+y

And taking this as

x=(x-y) + y.

When norms are used together with the triangle inequality, the result is:

|| x || = || x – y + y || ≤ || x – y || + || y ||

Hence proved.

Conclusion:

One of the characteristics that defines norms and measurements of distance is called the triangle inequality. This property must be established as a theorem for any function that is proposed for such purposes for each specific space. For instance, spaces such as the real numbers, Euclidean spaces, the  Lp spaces (p ≥ 1), and inner product spaces are examples of the types of spaces that fall under this category.