SECTRIX CURVE

A sectrix is an auxiliary curve allowing to graphically solve the problem of the section of an angle into n equal angles, with n any integer . The most famous case is the case n = 3, since it has no ruler-and-compass construction; the corresponding curves are called trisectrices

Examples of families of n-sectrix curves:
- the Maclaurin sectrices
- the Ceva sectrices.
- the Delanges sectrices.

Examples of n-sectrices for any n:
- the Archimedean spiral

Examples de trisectrices:
- the conchoid of Nicomedes
- the Maclaurin trisectrix
- the Tschirnhausen cubic (or Catalan trisectrix)
- the trisectrix limaçon
- the rectangular trefoil (or Longchamps trisectrix)
- the Ceva trisectrix
- the Delanges trisectrix
- the Dürer folium
- the hyperbola with eccentricity 2.

 Four of the nine previous trisectrices can be defined from the opposite figure, attributed to Archimedes, enclosing an angle and a triple angle. In the two following constructions, the length L is fixed.   If A and (D) are fixed, (D') rotates around A and C slides on (D), then the point D describes the Ceva trisectrix, with pole A (and the line (BC) envelopes an astroid). If C and D are fixed, (D) rotates around C and (D') rotates around D, then B can slide on (D') while describing a circle with centre C, but, most importantly, A describes the trisectrix limaçon, of pole D. In the 3 following constructions, the length L is variable. If A and C are fixed, and (D') rotates around A, then B describes the mediatrix of [AC], but most importantly, D describes the Maclaurin trisectrix, with pole A. If A and D are fixed, and (D) rotates around A, while B slides on (D'), then C describes a hyperbola with eccentricity 2. A fifth possibility, explored by Geneviève Tulloue, is fixing D and (D), rotating (D') around D, while A and C slide on (D). Then, the point B describes the curve with polar equation , that can be called "Tulloue trisectrix". AB = BC = CD=L

Animation of the construction of the Ceva trisectrix by this method (animations realised with the Cabri software by Geneviève Tulloue)

Animation of the construction of the trisectrix limaçon by this method:

Animation of the construction of the Maclaurin trisectrix:

Animation of the construction of the hyperbola with eccentricity 2 by this method:

Animation of the construction of the Tulloue trisectrix by this method: