Math 2.0

Geometry Foundations and axiom systems Theorem

Poncelet–Steiner theorem

Builds on Straightedge and compass constructionCircle

Statement

Let SS be a set of points in the Euclidean plane, and suppose that one circle and its centre are drawn in the plane. Every point that can be constructed from SS with straightedge and compass can also be constructed from SS and the given circle with straightedge alone.

Remarks

As for compass-only constructions, a circle counts as known when its centre and one of its points are known. The proof shows how to find, with straightedge alone and the help of the fixed circle, the intersection points of a line and a circle, and of two circles. Jean-Victor Poncelet stated the result in 1822, and Jakob Steiner proved it in 1833. The centre is needed: with straightedge alone one cannot find the centre of a single given circle. With straightedge alone and no circle, one obtains only points with coordinates in the field generated by the coordinates of SS. For example, 2\sqrt 2 cannot be constructed from 00 and 11 in this way.

Going deeper

For a student

The theorem and the Mohr–Mascheroni theorem show that the constructible points depend on the algebraic operations available, not on the tools themselves. Rational operations come from the straightedge, and square roots need some circle.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Circle, concept, no evidence levelCircleconceptStraightedge and compass construction, concept, no evidence levelStraightedge and comp…conceptPoncelet–Steiner theorem, statement, no evidence levelPoncelet–Steiner theo…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Circle concept
  2. Straightedge and compass construction concept
  3. Poncelet–Steiner theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Poncelet%E2%80%93Steiner_theorem
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AboutStraightedge and compass construction
UsesStraightedge and compass construction, Circle
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