<Dd> ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖, (\ displaystyle \ \ mathbf (x) + \ mathbf (y) \ \ leq \ \ mathbf (x) \ + \ \ mathbf (y) \,) </Dd> <P> where the length z of the third side has been replaced by the vector sum x + y . When x and y are real numbers, they can be viewed as vectors in R, and the triangle inequality expresses a relationship between absolute values . </P> <P> 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 proven without these theorems . The inequality can be viewed intuitively in either R or R . The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180 ° angle and two 0 ° angles, making the three vertices collinear, as shown in the bottom example . Thus, in Euclidean geometry, the shortest distance between two points is a straight line . </P> <P> 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 . </P>

When does the triangle inequality become an equality
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