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 ha, hb, hc; the medians ma, mb, mc; and the angle-bisectors wa, wb, wc.