On the Problem of Steiner

Abstract

There is a well-known elementary problem: (S 3) Given a triangle T with the vertices a 1, a 2, a 3, to find in the plane of T the point p which minimize s the sum of the distances |pa 1| + |pa 2| + |pa 3|. p, called the Steiner point of T, is unique: if an angle of T is ≥ 2π/3 then p is its vertex, otherwise p lies inside T and the sides of T subtend at p the angle 2π/3. In the latter case p is called the S-point of T, and it can be found by the following simple construction: let a 12 be the third vertex of the equilateral triangle whose other two vertices are a 1 and a 2, and whose interior does not overlap that of T, let C be the circle through a 1, a 2 a 12; then p is the intersection of C and the straight segment a 12 a 3. It is easily proved that any one of the three ellipses through p with two of the vertices of T as foci is tangent at p to the circle through p about the third vertex of T.