Betty Crocker Poundcakes

The problem of Steiner

The Steiner tree for n = 3 points
provides the minimum distance between points A, B, C (minimizes AS + BS + CS) and the point S Steiner point is called

The famous geometer Jacob Steiner, working in Berlin in the first half of the 800, dealt with numerous issues maximum and minimum by using different ways to determine isoperimetric properties of the circle and the sphere from which he deduced many applications.
One of the issues (Formerly known Caratheodory) shown by Steiner is as follows: Three villages A, B, C are to be joined by a road system of minimum total length. Mathematically the problem is reflected in the search, the plane in which lie the data points, a point P that is the minimum sum a + b + c, respectively, the distances of P from A, B and C. In the wake of the demonstration of the shear properties of the ellipse can be seen that the solution to the problem is this: If the triangle ABC all angles are less than 120 °, P is the point projecting each of the three sides AB; BC, AC, at an angle of 120 °. If an angle greater than or equal to 120 °, the point P coincides with the apex of the corner. The work of collection and processing was continued by R. Steiner Sturm in his book Maxima und Minima in der Geometrie elementaren 1910. The result of Steiner's most popular synthetic route is obtained by the theorem on isoperimetric , or that among all the gure plane of the circle is given perimeter that encloses the maximum area. Its synthetic methods were attacked by the analytical point of view of his contemporaries, first of all Dirichlet. Unfortunately, in fact, Steiner assumed maximizing the existence of the curve, while what is proved that if such a curve then there is a circle. The demonstration of a curve maximizing caused many problems to mathematicians in later years when fi Weierstrass did not appeal to the calculus of variations.