ZingPath: Triangle Side Lengths

The triangle inequality theorem for Euclidean geometry essentially states that the shortest distance between two points is a straight line. This intuitive concept is one of Euclid's axioms in his text Elements. Formally, the theorem can be stated as follows:

Triangle Inequality Theorem: In a triangle, one side length is less than the sum of the other two side lengths.

If a, b, and c are the lengths of the sides of a triangle, then the theorem gives the following inequalities:

While there are multiple proofs of the triangle inequality theorem, the proof Euclid used relies on is the side-angle relationship for triangles. This relationship states that if one angle of a triangle has greater measure than a second angle, then the side opposite the first is longer than the side opposite the second. The proof follows.

Let ABC be a triangle with side lengths a, b, and c. Construct point D so that and . See Figure 1. Now, is isosceles by construction, so . Since , which implies . By the side-angle relationship, this means that . Thus, . In a similar manner, we can show the other two inequalities.

So, the triangle inequality theorem provides an upper bound for the length of any side of a triangle. We can use this theorem to derive another theorem that will give us a lower bound.

Notice that because in Figure 1, it is also true that Since we also have that This means that and we have a lower bound for c. In a similar manner, we can find lower bounds for a and b. Formally, we have the following theorem:

Reverse Triangle Inequality Theorem: In a triangle, one side length is greater than the absolute value of the difference of the other two side lengths.

If a, b, and c are the lengths of the sides of a triangle, then the theorem gives the following inequalities:

Thus, the triangle inequality and the reverse triangle inequality give us a range for the length of a side of a triangle: