the triangle inequality

The triangle inequality is the geometric statement that the straight road is the shortest. For any triangle, the length of one side is always less than the sum of the other two — equivalently, going directly from one corner to another is never longer than detouring through the third corner. A side cannot be so long that the other two could not reach across to close the figure.

Precisely: in any triangle with side lengths a, b, c, all three of a < b + c, b < a + c, and c < a + b must hold. Turned around, three given lengths can form a genuine triangle if and only if the largest is strictly less than the sum of the other two — you only need to check the largest. The reason is the shortest-path idea: the straight segment between two points is shorter than any bent path, and a triangle's two short sides form exactly such a bent path between the endpoints of the long side. (Equality, side = sum of the other two, would flatten the triangle into a straight line — a degenerate, zero-area 'triangle'.)

This little inequality is everywhere: it certifies whether three sticks can make a triangle, it bounds distances, and abstracted to |x + y| <= |x| + |y| it becomes one of the defining axioms of distance in higher mathematics. A related fact in a single triangle: the longest side always lies opposite the largest angle, and the shortest opposite the smallest.