Definition
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.
Why It Matters
The triangle inequality is the ‘sanity check’ for geometry. It establishes the physical limits of connectivity: you cannot connect three points unless the two shorter paths are collectively longer than the longest one. This is a fundamental constraint in network design and pathfinding.
Core Concepts
- Fundamental Formula: For sides :
- How to read: “a plus b greater than c; a plus c greater than b; b plus c greater than a.”
- Meaning: Any two sides together must exceed the third. If equality holds, the three points are collinear, not a triangle.
- Exterior Angle Inequality: The measure of an exterior angle of a triangle is strictly greater than the measure of either nonadjacent (remote) interior angle.
- Hinge Theorem (SAS Inequality): If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is greater than the included angle of the second, then the third side of the first is longer than the third side of the second.
- Converse Hinge Theorem (SSS Inequality): If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is greater than the included angle of the second.
- Range of the Third Side: The length of any side () must be between the difference and the sum of the other two sides:
- How to read: “Absolute value of a minus b, less than c, less than a plus b.”
- Meaning: Given two sides, the third must be strictly between their difference and sum. Use this to find the allowable range for an unknown side.
- Side-Angle Inequality: In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.