Why any two sides of a triangle always outrun the third
In any triangle, two sides added together are at least as long as the third. Euclid proved it in Book I of the Elements, and the same rule now helps define what it means to measure distance at all, from ordinary space to abstract spaces of functions.
Euclid's Proposition 20 extends side AB past B to a point D so that BD equals BC, forming an isosceles triangle. Comparing angles shows AD must be longer than AC, and since AD is AB plus BC, those two sides together beat the third. Equality happens only in a degenerate triangle with zero area, where one angle is 180 degrees, the other two are zero and the three corners sit on one line; that is why the shortest path between two points in Euclidean geometry is a straight line.
For a proper triangle with sides a, b and c, the rule means three inequalities at once, which boil down to a single condition: the longest side must be shorter than the semiperimeter. Equivalently, Heron's formula must return a real, positive area. For right triangles it says the hypotenuse is longer than either leg but shorter than their sum. It follows from the Pythagorean theorem for right triangles and the law of cosines for others, though it can be proved without either.
In vector form, the length of a sum of two vectors never exceeds the sum of their lengths; for real numbers this becomes a statement about absolute values. The property is built into the definition of a norm or distance, and must be proved afresh for each candidate, whether on the real numbers, Euclidean spaces, Lp spaces with p at least 1, or inner product spaces.
On a sphere, where the shortest route is an arc of a great circle, the inequality still holds as long as distance is taken along the shorter arc between the two points.
Source: Triangle inequality
