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[Texts]## 51M15: Geometric constructions |

There is a theme in classical plane geometry in which one attempts to carry out various constructions using a ruler and compass. The simpler tasks are taught to us in primary and secondary school.

More general information about polygons (and polyhedra) is available on another page, as are a number of papers concerning plane geometry.

Parent field: 51M - Real and Complex Geometry

- Constructing the Geometric mean
- How to construct a hyperbola
- Construct the circle tangent to two lines and another circle
- For certain values of n one may construct a regular n-gon.
- Why are only certain n-gons constructible?
- Constructing a pentagon.
- Construction of the pentagon
- Constructing a heptagon
- Constructing the regular heptadecagon (17-sided polygon)
- Construction of the heptadecagon
- Summary and reference to construction of regular polygons and other classic problems
- Literature reviews: constructing the regular n-gon for n=257 or (allowing use of a trisector) for n=7, 13, 19, etc.
- Non-classical tools which allow trisection of angles
- What does it means to say one cannot square the circle?
- Non-constructibility of some extremal circle-points
- Given four points construct a square through them
- It turns out the constructions we are accustomed to can be carried out using only straightedge or only compass
- Steiner's Theorem: straightedge and compass constructions can be accomplished with straightedge, one circle, and its center; not without that center.
- Extending the Poncelet-Steiner ("no compass") theorem.
- Other variations on the Poncelet-Steiner ("no compass") theorem.
- The Mascheroni (no straightedge) theorem.
- Pointer to short proof of Mascheroni's theorem (no straightedges are needed for classical geometric constructions).
- A classic question is whether one can with compass and straightedge trisect an arbitrary angle; the answer is no, but there are ways to trisect angles with a
*marked*straightedge - Citation: Elementary proof that some angles cannot be trisected.
- Much easier! Trisecting a line segment
- Get your trisections here! and other unique mathematical experiences
- Trisecting an angle by motions off the plane.
- Apollonius' method of trisecting an angle.
- What can be constructed if we assume a trisector?
- Trisecting angles in the hyperbolic plane
- Using a quadratix to multisect an angle.
- Trisection is relevant if you wish to construct a regular nonagon (nine-sided polygon).
- Some things can still be done even with a short straightedge.
- Pointer to analysis of origami as an alternative to Euclidean constructions (origami allows more constructions, e.g. trisections).
- Pointer to paper on paper-folding
- Using projective geometry to perform a construction meeting incidence conditions.

Last modified 2000/01/14 by Dave Rusin. Mail: