3 - Inequalities and the existence of triangles

Summary

Since the time of Euclid, geometers have studied procedures for constructing triangles given elements such as the three sides, two sides and an angle, and so on. The construction procedure usually has a constraint, such as using only an unmarked straightedge and compass to draw the triangle. There are constraints on the given elements as well, usually given as inequalities. For example, as noted in Section 1.1, a triangle with sides of length a, b, and c can be constructed if and only if the three triangle inequalities a < b + c, b < c + a, and c < a + b hold.

In this chapter we examine inequalities among other elements of triangles that are necessary and sufficient for the existence of a triangle. We discuss the altitudes h_a, h_b, h_c; the medians m_a, m_b, m_c; and the angle-bisectors w_a, w_b, w_c.

Inequalities in the Elements of Euclid

In the first book of the Elements, Euclid states sixteen properties of equality and inequality. Of his five Common Notions, just one, the fifth, deals with inequality: The whole is greater than the part. Other properties of inequality appear in proofs of propositions. For example, in the proof of Proposition I.17 (In any triangle, the sum of any two angles is less than two right angles), Euclid uses the property that if x < y, then x + z < y + z [Joyce].