A simple, common, but very powerful tool for illustrating inequalities among positive numbers is to represent such numbers by means of line segments whose lengths are the given positive numbers. In this chapter, and in the ones to follow, we illustrate inequalities by comparing the lengths of the segments using one or more of the following methods:

1. The inclusion principle. Show that one segment is a subset of another. We will generalize this method in the next chapter when we represent numbers by areas and volumes and illustrate inequalities via the subset relation.

2. The geodesic principle. Use the well-known fact that the shortest path joining two points is the linear segment connecting them.

3. The Pythagorean comparison. Proposition I.19 in The Elements of Euclid states “In any triangle the side opposite the greater angle is greater.” Thus in a right triangle, the hypotenuse is always the longest side. So to compare two segments, show that one is a leg and the other the hypotenuse of a right triangle.

4. The triangle (and polygon) inequality. Proposition I.20 in The Elementsm states “In any triangle the sum of any two sides is greater than the remaining one.” Thus when three line segments form a triangle, the length of any one of them is less than or equal to the sum of the other two (and similarly for polygons). This is a special case of the geodesic principle.

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