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Forum Geometricorum Volume 1 (2001) 7–8.

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FORUM GEOM

Another Proof of the Erd˝os-Mordell Theorem Hojoo Lee Abstract. We give a proof of the famous Erd˝os-Mordell inequality using Ptolemy’s theorem.

The following neat inequality is well-known: Theorem. If from a point O inside a given triangle ABC perpendiculars OD, OE, OF are drawn to its sides, then OA + OB + OC ≥ 2(OD + OE + OF ). Equality holds if and only if triangle ABC is equilateral. A

E

F O B

D

C

Figure 1

This was conjectured by Paul Erd˝os in 1935, and first proved by Louis Mordell in the same year. Several proofs of this inequality have been given, using Ptolemy’s theorem by Andr´e Avez [5], angular computations with similar triangles by Leon Bankoff [2], area inequality by V. Komornik [6], or using trigonometry by Mordell and Barrow [1]. The purpose of this note is to give another elementary proof using Ptolemy’s theorem. Proof. Let HG denote the orthogonal projections of BC on the line F E. See Figure 2. Then, we have BC ≥ HG = HF + F E + EG. It follows from ∠BF H = ∠AF E = ∠AOE that the right triangles BF H and AOE are similar OF OE BF . In a like manner we find that EG = CE. Ptolemy’s and HF = OA OA theorem applied to AF OE gives AF · OE + AE · OF . OA · F E = AF · OE + AE · OF or F E = OA Combining these, we have BC ≥

AF · OE + AE · OF OF OE BF + + CE, OA OA OA

Publication Date: January 29, 2001. Communicating Editor: Paul Yiu.

8

H. Lee

A

H

E

F

G

O B

D

C

Figure 2

or BC · OA ≥ OE · BF + AF · OE + AE · OF + OF · CE = OE · AB + OF · AC. AC AB OE + OF . Dividing by BC, we have OA ≥ BC BC Applying the same reasoning to other projections, we have BA CA CB BC OF + OD and OC ≥ OD + OE. OB ≥ CA CA AB AB Adding these inequalities, we have AB CB AC BC BA CA + )OD + ( + )OE + ( + )OF. OA + OB + OC ≥ ( CA AB BC AB BC CA x y It follows from this and the inequality + ≥ 2 (for positive real numbers x, y x y) that OA + OB + OC ≥ 2(OD + OE + OF ). It is easy to check that equality holds if and only if AB = BC = CA and O is the circumcenter of ABC.  References [1] P. Erd˝os, L. J. Mordell, and D. F. Barrow, Problem 3740, Amer. Math. Monthly, 42 (1935) 396; solutions, ibid., 44 (1937) 252 – 254. [2] L. Bankoff, An elementary proof of the Erd˝os-Mordell theorem, Amer. Math. Monthly, 65 (1958) 521. [3] A. Oppenheim, The Erd˝os inequality and other inequalities for a triangle, Amer. Math. Monthly, 68 (1961), 226 - 230. [4] L. Carlitz, Some inequalities for a triangle, Amer. Math. Monthly, 71 (1964) 881 – 885. [5] A. Avez, A short proof of a theorem of Erd˝os and Mordell, Amer. Math. Monthly, 100 (1993) 60 – 62. [6] V. Komornik, A short proof of the Erd˝os-Mordell theorem, Amer. Math. Monthly, 104 (1997) 57 – 60. Hojoo Lee: Department of Mathematics, Kwangwoon University, Wolgye-Dong, Nowon-Gu, Seoul 139-701, Korea E-mail address: [email protected]

Another Proof of the Erd˝os-Mordell Theorem

Jan 29, 2001 - This was conjectured by Paul Erd˝os in 1935, and first proved by Louis Mordell in the same year. Several proofs of this inequality have been given, using Ptolemy's theorem by André Avez [5], angular computations with similar triangles by Leon. Bankoff [2], area inequality by V. Komornik [6], or using ...
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