Tomahawk (geometric Shape) - Trisection

Trisection

To use the tomahawk to trisect an angle, it is placed with its handle line touching the apex of the angle, with the blade inside the angle, tangent to one of the two rays forming the angle, and with the spike touching the other ray of the angle. One of the two trisecting lines then lies on the handle segment, and the other passes through the center point of the semicircle. If the angle to be trisected is too sharp relative to the length of the tomahawk's handle, it may not be possible to fit the tomahawk into the angle in this way, but this difficulty may be worked around by repeatedly doubling the angle until it is large enough for the tomahawk to trisect it, and then repeatedly bisecting the trisected angle the same number of times as the original angle was doubled.

If the apex of the angle is labeled A, the point of tangency of the blade is B, the center of the semicircle is C, the top of the handle is D, and the spike is E, then triangles ACD and ADE are both right triangles with a shared base and equal height, so they are congruent triangles. Because the sides AB and BC of triangle ABC are respectively a tangent and a radius of the semicircle, they are at right angles to each other and ABC is also a right triangle; it has the same hypotenuse as ACD and the same side lengths BC = CD, so again it is congruent to the other two triangles, showing that the three angles formed at the apex are equal.

Although the tomahawk may itself be constructed using a compass and straightedge, and may be used to trisect an angle, it does not contradict Pierre Wantzel's 1837 theorem that arbitrary angles cannot be trisected by compass and unmarked straightedge alone. The reason for this is that placing the constructed tomahawk into the required position is a form of neusis that is not allowed in compass and straightedge constructions.