Triangle inequality

The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom).

Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints.

[4][5] The triangle inequality is a defining property of norms and measures of distance.

Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure.

[7] For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths a, b, c that are all positive and excludes the degenerate case of zero area): A more succinct form of this inequality system can be shown to be Another way to state it is implying and thus that the longest side length is less than the semiperimeter.

A mathematically equivalent formulation is that the area of a triangle with sides a, b, c must be a real number greater than zero.

An isosceles triangle ABC is constructed with equal sides AB = AC.

Hence: A similar construction shows AC > DC, establishing the theorem.

This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides.

This now leaves the first and third inequalities needing to satisfy The first of these quadratic inequalities requires r to range in the region beyond the value of the positive root of the quadratic equation r2 + r − 1 = 0, i.e. r > φ − 1 where φ is the golden ratio.

The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints.

This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line.

This implies that no curve can have an arc length less than the distance between its endpoints.

In either case, if the side lengths are a, b, c we can attempt to place a triangle in the Euclidean plane as shown in the diagram.

and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so

For example, the triangle inequality appears to allow the possibility of four points A, B, C, and Z in Euclidean space such that distances and However, points with such distances cannot exist: the area of the 26–26–26 equilateral triangle ABC is

, the area of a 26–14–14 isosceles triangle (all by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality.

[15] If the normed space is Euclidean, or, more generally, strictly convex, then

For instance, consider the plane with the ℓ1 norm (the Manhattan distance) and denote u = (1, 0) and v = (0, 1).

Then the triangle formed by u, v, and u + v, is non-degenerate but The absolute value is a norm for the real line; as required, the absolute value satisfies the triangle inequality for any real numbers u and v. If u and v have the same sign or either of them is zero, then

There is also a lower estimate, which can be found using the reverse triangle inequality which states that for any real numbers u and v,

The taxicab norm or 1-norm is one generalization absolute value to higher dimensions.

Taking the square root of the final result gives the triangle inequality.

The p-norm is a generalization of taxicab and Euclidean norms, using an arbitrary positive integer exponent,

where the vi are the components of vector v. Except for the case p = 2, the p-norm is not an inner product norm, because it does not satisfy the parallelogram law.

For metric spaces, the proof of the reverse triangle inequality is found similarly by:

By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that[20]

Moreover, if u and v are both timelike vectors lying in the future light cone, the triangle inequality is reversed: A physical example of this inequality is the twin paradox in special relativity.

is space-like (and therefore a Euclidean subspace) then the usual triangle inequality holds.