trisecting

Trisecting is the process of dividing an object into three equal parts. In geometry, it most often refers to trisecting a line segment or an angle. Trisecting a segment is a standard constructive task in Euclidean geometry, while trisecting an arbitrary angle is one of the well-known classical problems about what can be achieved with straightedge and compass.

Trisecting a line segment: Given a segment AB, there are straightforward straightedge-and-compass methods to locate two points that divide AB into three congruent parts. A typical approach uses a triangle and lines parallel to AB to create three equal subsegments. The constructions rely on the properties of similar triangles and are routine in geometric drawing.

Trisecting an angle: The angle trisection problem asks for a general method to divide any given angle into three equal angles using only a straightedge and compass. It is not possible in general: there exist angles that cannot be trisected with those tools. The obstruction arises from the algebraic nature of the triple-angle equation and the corresponding limitations identified by Galois theory, which show that solving for a trisection does not always reduce to a sequence of quadratic constructions. Nevertheless, some specific angles can be trisected, and alternative techniques such as neusis constructions or origami can trisect angles that are not solvable by traditional methods.

Other uses: Trisecting can also refer to partitioning other geometric objects or regions into three equal parts, or dividing a curve, polygon edge, or path into three equal-length segments. The term is commonly applied in drafting, design, and geometric reasoning.