Lesson plan
Trisectrix
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 * Tschirnhausen cubic (a.k.a. Catalan's trisectrix and L'Hôpital's cubic) * Deslanges trisectrix * 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
Article summary
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 * Tschirnhausen cubic (a.k.a. Catalan's trisectrix and L'Hôpital's cubic) * Deslanges trisectrix * 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
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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