tangency of the blade is , the center of the semicircle is , the top of the handle is , and the spike is , then triangles and are both right triangles with a shared base and equal height, so they are congruent triangles. Because the sides and of triangle are respectively a tangent and a radius of the semicircle, they are at right angles to each other and is also a right triangle; it has the same hypotenuse as and the same side lengths , 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. History The inventor of the tomahawk is unknown, but the earliest references to it come from 19th-century France. It dates back at least as far as 1835, when it appeared in a book by Claude Lucien Bergery, Géométrie appliquée à l'industrie, à l'usage des artistes et des ouvriers (3rd edition). Another early publication of the same trisection was made by Henri Brocard in 1877; Brocard in turn attributes its invention to an 1863 memoir by French naval officer . References External links Trisection using special tools: "Tomahawk", Takaya Iwamoto, 2006, featuring a tomahawk tool made from transparent vinyl and comparisons for accuracy against other trisectors Construction heptagon with tomahawk, animation Mathematical tools