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On the Problem of Steiner

Published online by Cambridge University Press:  20 November 2018

Z.A. Melzak
Z.A. Melzak*
Affiliation:
University of British Columbia
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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.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 4 , Issue 2 , May 1961 , pp. 143 - 148
Copyright
Copyright © Canadian Mathematical Society 1961

References

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