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.
Atlantic slave trade The first side of the triangle was the export of goods from Europe to Africa. A number of African kings and merchants took part in the trading of enslaved people from 1440 to about 1833. For each captive, the African rulers would receive a variety of goods from Europe. These included guns, ammunition, and other factory-made goods. The second leg of the triangle exported enslaved Africans across the Atlantic Ocean to the Americas and the Caribbean Islands. The third and final part of the triangle was the return of goods to Europe from the Americas. The goods were the products of slave-labour plantations and included cotton, sugar, tobacco, molasses and rum.[47] Sir John Hawkins, considered the pioneer of the British slave trade, was the first to run the Triangular trade, making a profit at every stop.