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How does the triangle inequality determine whether three lengths can form a triangle?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In Euclidean geometry, the triangle inequality serves as the fundamental rule for determining whether three given lengths can form a triangle.

For three line segments with lengths a, b, and c to successfully form a triangle, the sum of any two sides must be strictly greater than the third side. This requirement is expressed by the following three conditions:1

a < b + c
b < a + c
c < a + b

If the sum of any two sides is exactly equal to the third side (e.g., a + b = c), the segments do not form a standard triangle; instead, they create a "degenerate triangle," where all three vertices lie on a single straight line. If the sum of two sides is less than the third, the segments cannot meet to close the shape, meaning no triangle can be formed. In essence, this theorem reflects the geometric principle that the shortest distance between two points is a straight line.21