Monge's theorem (elementary geometry)

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Monge's theorem
The three outer centers of extension lie on a common (red) straight line.

The set of Monge is a theorem of elementary geometry , which the French mathematician Gaspard Monge back. The theorem deals with a property of circles of the Euclidean plane in connection with centric elongations .

Formulation of the sentence

The sentence can be stated as follows:

For three pairs of separated circles of the Euclidean plane with different radii , which are converted into one another by three centric extensions , the three outer extension centers are always located on a straight line .

Explanations

  1. In the Euclidean plane two circles are separated when the associated circular disks are disjoint .
  2. In the Euclidean plane, the outer center of extension of two separate circles with different radii is obtained as the intersection of the two outer circle tangents . This point of intersection is therefore not on the line connecting the two circle centers .

Historical note

literature

Web link

Footnotes and individual references

  1. ^ Theophil Lambacher, Wilhelm Schweizer: Geometry. Edition E. Part 2. 1965, p. 152
  2. ^ Johannes Kratz, Karl Wörle: Geometry. Part II. 1968, p. 66
  3. ↑ a b David Wells: The Penguin Dictionary of Curious and Interesting Geometry 1991, pp. 153-154
  4. ↑ A point of similarity is also used instead of the center of extension .