An outscribed triangle of a triangle 4ABC is a triangle 4DEF such that each side of 4DEF contains a vertex of 4ABC. In this article we study the equilateral outscribed triangles of an arbitrary triangle and determine the area of the largest such triangles. We prove that the largest outscribed equilateral triangle of 4ABC can be constructed by ruler and compass and its area equals a +b+c 2 √ 3 +2S4ABC where S4ABC denotes the area of 4ABC. Given two triangles 4ABC and 4DEF , if each side of 4DEF contains a vertex of 4ABC, then we call 4DEF an outscribed triangle of 4ABC. Given 4ABC, let Φ4ABC be the set of all outscibed equilateral triangles of 4ABC. Clearly Φ4ABC is non-empty. In the following we will determine the area of the largest member of Φ4ABC and show that this largest member can be constructed by ruler and compass from 4ABC. The corresponding problem on quadrilaterals has been considered in [1]. 1. Area of the largest outscribed equilateral triangle Given a triangle 4ABC, let a and b denote the lengths of the sides BC and AC, respectively, and θ denote the angle ∠ACB. Let 4DEF be any member in Φ4ABC as shown in Figure 1 and put t = ∠DCB.