Trisectrix

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title: "Trisectrix" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["curves"] description: "Curve which could be used to trisect an angle with compass and straightedge" topic_path: "general/curves" source: "https://en.wikipedia.org/wiki/Trisectrix" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Curve which could be used to trisect an angle with compass and straightedge ::

In geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle with ruler and compass and this curve as an additional tool. Such a method falls outside those allowed by compass and straightedge constructions, so they do not contradict the well known theorem which states that an arbitrary angle cannot be trisected with that type of construction. There is a variety of such curves and the methods used to construct an angle trisector differ according to the curve. Examples include:

• Limaçon trisectrix (some sources refer to this curve as simply the trisectrix.)
• Trisectrix of Maclaurin{{citation | last = Dudley | first = Underwood | author-link = Underwood Dudley | isbn = 0883855143 | page = 12 | publisher = Cambridge University Press | title = The Trisectors | year = 1994}};
• Tschirnhausen cubic (a.k.a. Catalan's trisectrix and L'Hôpital's cubic){{citation | last = Farouki | first = Rida T. | doi = 10.1007/978-3-540-73398-0 | isbn = 978-3-540-73397-3 | mr = 2365013 | pages = 398–399 | publisher = Springer | series = Geometry and Computing | title = Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable | volume = 1 | year = 2008}}
• Cubic parabola (the graph of the cube function)
• Hyperbola with eccentricity 2
• Parabola
• Cycloid of Ceva

A related concept is a sectrix, which is a curve which can be used to divide an arbitrary angle by any integer. Examples include:

• Archimedean spiral
• Quadratrix of Hippias
• Sine curve{{citation | last = Sheng | first = Hung Tao | doi = 10.1080/0025570X.1969.11975925 | journal = Mathematics Magazine | jstor = 2689193 | mr = 240707 | pages = 73–80 | title = A method of trisection of an angle and X-section of an angle | volume = 42 | year = 1969 | issue = 2

References

References

1. {{EB1911
2. Yates, Robert C.. (January 1941). "The trisection problem, chapter II: Solutions by means of curves". National Mathematics Magazine.
3. Wright, J. M. F.. (1836). "An Algebraic System of Conic Sections, and Other Curves". Black and Armstrong.
4. Ferréol, Robert. (2017). "Sectrix curve". Encyclopédie des formes mathématiques remarquables.
5. (2011). "A History of Mathematics". John Wiley & Sons.
6. (July 2006). "An investigation of historical geometric constructions". Mathematical Association of America.

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