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.
Equality before the law Equality before the law, also known as: equality under the law, equality in the eyes of the law, or legal equality, is the principle that each independent human being must be treated equally by the law (principle of isonomy) and that all people are subject to the same laws of justice (due process).[1] Therefore, the law must guarantee that no individual nor group of individuals should be privileged or discriminated against by the government. Equality before the law is one of the basic principles of liberalism.[2][3] This principle arises from various important and complex questions concerning equality, fairness, and justice. In 1894, the author Anatole France said that "[i]n its majestic equality, the law forbids rich and poor alike to sleep under bridges, beg in the streets, and steal loaves of bread."[4] The belief in equality before the law is called legal egalitarianism. The principle of equality before the law is incompatible and ceases to exist with legal systems such as slavery, servitude, colonialism, monarchy, or quotaism.