page_title: Angle trisection
section_title: With a "tomahawk"
hierarchical_section_title: Angle trisection / Other methods / With a "tomahawk"
context_page_description: Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
  The problem as stated is impossible to solve for arbitrary angles, as proved by Pierre Wantzel in 1837. However, although there is no way to trisect an angle in general with just a compass and a straightedge, some special angles can be trisected. For example, it is relatively straightforward to trisect a right angle.
  It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries.
  Because it is defined in simple terms, but complex to prove unsolvable, the problem of angle trisection is a frequent subject of pseudomathematical attempts at solution by naive enthusiasts.
context_section_description: A "tomahawk" is a geometric shape consisting of a semicircle and two orthogonal line segments, such that the length of the shorter segment is equal to the circle radius. Trisection is executed by leaning the end of the tomahawk's shorter segment on one ray, the circle's edge on the other, so that the "handle" (longer segment) crosses the angle's vertex; the trisection line runs between the vertex and the center of the semicircle.
  Note that while a tomahawk is constructible with compass and straightedge, it is not generally possible to construct a tomahawk in any desired position.  Thus, the above construction does not contradict the nontrisectibility of angles with ruler and compass alone.
  The tomahawk produces the same geometric effect as the paper-folding method: the distance between circle center and the tip of the shorter segment is twice the distance of the radius, which is guaranteed to contact the angle.  It is also equivalent to the use of an architects L-Ruler (Carpenter's Square).
caption: A tomahawk trisecting an angle. The handle forms one trisector and the blue line shown forms the other.