Poncelet–Steiner theorem

File:Steiner construction of a parallel to a diameter.gif

To draw the parallel (h) to a diameter g through any given point P. Chose auxiliary point C anywhere on the straight line through B and P outside of BP. (Steiner)

In Euclidean geometry, the Poncelet–Steiner theorem concerning compass and straightedge constructions states that whatever can be constructed by straightedge and compass together can be constructed by straightedge alone, provided that a single circle and its centre are given. This result can not be weakened; if the centre of the circle is not given, it cannot be constructed by a straightedge alone. Also, the entire circle is not required. In 1904, Francesco Severi proved that any small arc together with the centre will suffice.

The result was conjectured by Jean Victor Poncelet in 1822, and proven by Jakob Steiner in 1833.

See also

• Mohr–Mascheroni theorem, the theorem that any compass and straightedge construction can be performed with only a compass

External links

• Jacob Steiner's theorem at cut-the-knot (It is impossible to find the center of a given circle with the straightedge alone)
• Straightedge alone Basic constructions of straightedge-only constructions.