Tomahawk (drawing device)

tomahawk

The tomahawk is a geometric "tool" that consists of a semicircle and two straight lines. The basic shape can be constructed with a compass and ruler . With a tomahawk you can divide an angle into three . This does not contradict the insolubility of the classic problem , since the required displacement and rotation of the constructed tomahawk into the required position goes beyond the construction methods permitted there.

Angular trisection

Tripartite an angle with a tomahawk

Place the tomahawk at a given angle so that its handle goes through the tip of the angle, its blade touches one leg of the angle, and its rear end touches the other leg. Now the handle forms two angles with the two legs, the dimensions of which are a third or two thirds of the starting angle.

Due to the construction of the tomahawk, the CD and DE are the same length and the ADC and ADE angles are right angles. According to the congruence theorem SWS , the triangles ADC and ADE are thus congruent. Furthermore, the triangle ABC is also congruent to these two, since it has a right angle in B ( circle tangent and radius ), the side lengths of BC and CD match (radius) and it has the side AC in common with the triangle ADC . Because of the congruence of the three triangles, the three angles BAC , CAD and DAE are equal and thus divide the given angle BAE into three equal parts.

history

It is not certain who invented the tomahawk. The earliest known mentions are from the 19th century in France. There he is mentioned in the book Géométrie appliquée a l'industrie, a l'usage des artistes et des ouvriers by Claude Lucien Bergery, published in 1835. Henri Brocard described it in 1877 in a publication in which he attributed the discovery to the naval officer Pierre-Joseph Glotin, who had treated it in 1863 in an essay for the Mémoires de la Société des sciences physiques et naturelles de Bordeaux .

Web links

Commons : Tomahawk (geometry) - collection of images, videos and audio files

MathWorld / Wolfram (English)
Angle Trisection problem: trisect An Angle With a Tomahawk (English)

Individual evidence

↑ C. Stanley Olgivy: Excursions in Geometry . Dover 1990, ISBN 0-486-26530-7 , pp. 139-140 .
^ Yates, Robert C. "The Trisection Problem" p. 37 Fig. 21, THE FRANKLIN PRESS, INC. BATON ROUGE, LA 1942, 68 pages, accessed December 3, 2015.
↑ Underwood Dudley : The Trisectors. 2nd Edition. MAA Spectrum, 1994, ISBN 0-88385-514-3 , pp. 14-16 (Note: Dudley erroneously writes Bricard instead of Brocard and Glatin instead of Glotin).
^ Henri Brocard : Note sur la division mécanique de l'angle . In: Bulletin de la Société Mathématique de France. 5, pp. 43-47 (1877).
↑ M. Glotin: De quelques moyens pratiques de diviser les angles de parties égales. In: Mémoires de la Société des Sciences physiques et naturelles de Bordeaux. Volume 2, pp. 253-278 .

[1] C. Stanley Olgivy: Excursions in Geometry . Dover 1990, ISBN 0-486-26530-7 , pp. 139-140 .

[2] Yates, Robert C. "The Trisection Problem" p. 37 Fig. 21, THE FRANKLIN PRESS, INC. BATON ROUGE, LA 1942, 68 pages, accessed December 3, 2015.

[3] Underwood Dudley : The Trisectors. 2nd Edition. MAA Spectrum, 1994, ISBN 0-88385-514-3 , pp. 14-16 (Note: Dudley erroneously writes Bricard instead of Brocard and Glatin instead of Glotin).

[4] Henri Brocard : Note sur la division mécanique de l'angle . In: Bulletin de la Société Mathématique de France. 5, pp. 43-47 (1877).

[5] M. Glotin: De quelques moyens pratiques de diviser les angles de parties égales. In: Mémoires de la Société des Sciences physiques et naturelles de Bordeaux. Volume 2, pp. 253-278 .