On the Problem of Steiner
Canadian mathematical bulletin, Tome 4 (1961) no. 2, pp. 143-148
Voir la notice de l'article provenant de la source Cambridge University Press
There is a well-known elementary problem:(S3) Given a triangle T with the vertices a1, a2, a3, to find in the plane of T the point p which minimize s the sum of the distances |pa1| + |pa2| + |pa3|.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 a12 be the third vertex of the equilateral triangle whose other two vertices are a1 and a2, and whose interior does not overlap that of T, let C be the circle through a1, a2 a12; then p is the intersection of C and the straight segment a12a3. 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.
Melzak, Z.A. On the Problem of Steiner. Canadian mathematical bulletin, Tome 4 (1961) no. 2, pp. 143-148. doi: 10.4153/CMB-1961-016-2
@article{10_4153_CMB_1961_016_2,
author = {Melzak, Z.A.},
title = {On the {Problem} of {Steiner}},
journal = {Canadian mathematical bulletin},
pages = {143--148},
year = {1961},
volume = {4},
number = {2},
doi = {10.4153/CMB-1961-016-2},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1961-016-2/}
}
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