Forum Geometricorum Volume 5 (2005) 143–148. FORUM GEOM ISSN 1534-1178 Applications of Homogeneous Functions to Geometric Inequalities and Identities in the Euclidean Plane Wladimir G. Boskoff and Bogdan D. Suceav˘ a Abstract. We study a class of geometric identities and inequalities that have a common pattern: they are generated by a homogeneous function. We show how to extend some of these homogeneous relations in the geometry of triangle. Then, we study the geometric configuration created by two intersecting lines and a pencil of n lines, where the repeated use of Menelaus’s Theorem allows us to emphasize a result on homogeneous functions. 1. Introduction The purpose of this note is to present an extension of a certain class of geometric identities or inequalities. The idea of this technique is inspired by the study of homogeneous polynomials and has the potential for additional applications besides the ones described here. First of all, we recall that a function f : R n → R is called homogeneous if f (tx 1 , tx 2 , ..., tx n )= t m f (x 1 ,x 2 , ...x n ), for t ∈ R -{0} and x i ∈ R,i =1, ..., n, m, n ∈ N,m =0,n ≥ 2. The natural number m is called the degree of the homogeneous function f. Remarks. 1. Let f : R n → R be a homogeneous function. If for x =(x 1 , ..., x n ) ∈ R n , we have f (x) ≥ 0, then f (tx) ≥ 0, for t> 0. Furthermore, if m is an even natural number, f (x) ≥ 0, yields f (tx) ≥ 0 for any real number t. 2. Any x> 0 can be written as x = a b , with a, b ∈ (0, 1). 2. Application to the geometry of triangle Consider the homogeneous function f α : R 3 → R given by f α (x 1 ,x 2 ,x 3 )= αx 1 x 2 x 3 , with α ∈ R -{0}. Denote by a, b, c the lengths of the sides of a triangle ABC, by R the circumradius and by △ the area of this triangle. By the law of sines, we get f 1 (a, b, c)= f 1 (a, b, 2R sin C )=2Rf 1 (a, b, sin C )=4R△. Thus, we obtain abc =4R△. Publication Date: October 11, 2005. Communicating Editor: Paul Yiu.