A Visual Proof of the Erdős-Mordell Inequality

We present a visual proof of a lemma that reduces the proof of the Erdős-Mordell inequality to elementary algebra. In 1935, the following problem proposal appeared in the “Advanced Problems” section of theAmerican Mathematical Monthly [5]: 3740. Proposed by Paul Erdős, The University, Manchester, England. From a pointO inside a given triangleABC the perpendiculars OP , OQ, OR are drawn to its sides. Prove that OA + OB + OC ≥ 2(OP + OQ + OR). Trigonometric solutions by Mordell and Barrow appeared in [11]. The proofs, however, were not elementary. In fact, no “simple and elementary” proof of what had become known as the Erd ̋ os-Mordell theorem was known as late as 1956 [13]. Since then a variety of proofs have appeared, each one in some sense simpler or more elementary than the preceding ones. In 1957 Kazarinoff published a proof [7] based upon a theorem in Pappus of Alexandria’s Mathematical Collection; and a year later Bankoff published a proof [2] using orthogonal projections and similar triangles. Proofs using area inequalities appeared in 1997 and 2004 [4, 9]. Proofs employing Ptolemy’s theorem appeared in 1993 and 2001 [1, 10]. A trigonometric proof of a generalization of the inequality in 2001 [3], subsequently generalized in 2004 [6]. Many of these authors speak glowingly of this result, referring to it as a “beautiful inequality” [9], a “remarkable inequality” [12], “the famous Erd ̋ osMordell inequality” [4, 6, 10], and “the celebrated Erd ̋ os-Mordell inequality. . . a beautiful piece of elementary mathematics” [3]. In this short note we continue the progression towards simpler proofs. First we present a visual proof of a lemma that reduces the proof of the Erd ̋ os-Mordell inequality to elementary algebra. The lemma provides three inequalities relating the lengths of the sides of ABC and the distances fromO to the vertices and to the sides. While the inequalities in the lemma are not new, we believe our proof of the lemma is. The proof uses nothing more sophisticated than elementary properties of triangles. In Figure 1(a) we see the triangle as described by Erd ̋ os, and in Figure Publication Date: April 30, 2007. Communicating Editor: Paul Yiu. 100 C. Alsina and R. B. Nelsen 1(b) we denote the lengths of relevant line segments by lower case letters, whose use will simplify the presentation to follow. In terms of that notation, the Erd ̋ osMordell inequality becomes x + y + z ≥ 2(p + q + r).