### Poncelet-Steiner Theorem

The Poncelet-Steiner theorem says

Everything you can construct with a straightedge and a compass you can construct by the straightedge alone, provided you are given a circle and its center.

Motivated by Mascheroni’s result ↑ J.V.Poncelet conjectured this results in 1812  [1]  and it was proved by J.Steiner [2]  in 1833.

It can be shown that the constructions cannot be done by straightedge alone [3] . By the straightedge alone only the so called linear constructions can be done. For instance, using the straightedge alone, without a circle given, is not sufficient to construct square roots. Even simpler constructions as to half a straight line segment are impossible by the straightedge alone. Another example is the result known as Steiner’s theorem:

Steiner’s Theorem: It is impossible to find the center of a given circle with the straightedge alone.

The basic idea of the following proof goes back to Hilbert. If such a construction would be possible, then it would be preserved by projective transformations.  This due to the basic properties of projective transformation which preserve lines, objects constructible by the straightedge. On the other hand, the circle as a conic section is transformed to a conic section in general.  Even worse, the conjugate diameters 1 of a conic section pre-image may not be transformed to the conjugate diameters of the image. Consequently, the center of circle is not projected to the center of the image.

It can be shown using only elementary projective geometry that also the center of the circle in Steiner’s result is indispensable. Even two given and not intersecting circles with unknown centers do no suffice. On the other hand, two intersecting circles with unknown center suffice. The same is true for three not intersecting circles with unknown centers [3] .

According to M. Chasles  (1793 - 1880) [4]  already Schooten (died 1659) gave first constructions using the straightedge as soon as some other means (such as two parallel lines, halved segment etc.) are given on the drawing-paper  Some notes on the role of the straightedge constructions in the perspective and geometry can also  be found in [5] .

E.Knabl [6] proved that the circle can be replaced by a conic section with its foci (or with the center and one focus) in the Poncelet-Steiner theorem.

### Notes

 1 Any chord through the center of an ellipse  specifies a family of parallel chords that includes the tangents of the same slope. The mid-points of all these chords define another chord, that is the diameter conjugate to the original diameter. A special case are the of the conjugate diameters are  the major and minor axes. Properties that hold relative to the perpendicular axes also hold with respect to the conjugate diameters.

### References

 [1] Poncelet, J. V. (1822). Traité des propriétés projectives des figures. Paris.

 [2] Steiner, J. (1933). Die geometrischen Konstruktionen, ausgeführt mittels der geraden Linie und eines festen Kreises, als Lehrgegenstand auf höheren Unterrichts-Lehranstalten und zur praktischen Benutzung (Geometrical Constructions Using a Straight Line and a Fixed Circle). Berlin: (cf. Oswalds Klassiker der exackten Wissenschaft, Nr. 60).

 [3] Rademacher, H., & Toeplitz, O. (1933). Von Zahlen und Figuren (2nd ed.). Berlin: Springer Verlag.

 [4] Chasles, M. (1837). Aperçu historique sur l'origine et le développement des méthodes en géométrie (Historical view of the origin and development of methods in geometry). Bruxelles: M.Hayez.

 [5] Lambert, J. H. (1759). Freye Perspective - oder Anweisung, jeden perspektivischen Aufriß von freyen Stücken und ohne Grundriß zu verfertigen. Zürich.

 [6] Knabl, E. (1881). Die geometrische Konstruktionen der Aufgaben 1. und 2. Grades. Annual report of the grammar-school, Salzburg.

Cite this web-page as:

Štefan Porubský: Poncelet-Steiner Theorem.

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