when does the triangle inequality become an equality
Triangle inequality 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 ℝ2 or ℝ3. 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.
Law of cosines The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle γ is a right angle (of measure 90° [degrees], or π/2 radians), then cos γ = 0, and thus the law of cosines reduces to the Pythagorean theorem: