What is triangle inequality?

The Triangle Inequality is a fundamental concept in geometry stating that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. This principle holds true for all types of triangles.

More formally, let a, b, and c be the lengths of the sides of a triangle. Then, the following three inequalities must be satisfied:

• a + b > c
• a + c > b
• b + c > a

If any of these inequalities is not satisfied, then it is impossible to construct a triangle with those side lengths.

The Triangle Inequality Theorem is used extensively in various mathematical fields, including:

• Determining if three given side lengths can form a triangle.
• Proving other geometric theorems.
• Solving problems involving distances and shortest paths.

A useful application can be found in distance calculations. The shortest distance between two points is always a straight line; hence, any detour will always be longer. You can think of the direct path between two points as one side of a hypothetical triangle, and the detour as the sum of the other two sides, therefore following the rule.

Related concepts include:

• Degenerate Triangles: A degenerate triangle occurs when the sum of two sides is equal to the third side (a + b = c). In this case, the "triangle" collapses into a straight line. https://www.wikiwhat.page/kavramlar/Degenerate%20Triangles

• Euclidean Distance: The triangle inequality is foundational to the concept of Euclidean distance and ensures that distances are well-defined. https://www.wikiwhat.page/kavramlar/Euclidean%20Distance