Historically, a contrived trisected line was used to trisect any other line, using the principle of projection. This is in essence about relationship and its accomplishment is about working backwards. Loosely speaking, any angle comprises two connecting lines. Attempts at trisecting any angle, which is dividing it into three equal parts, failed. In this paper any angle is defined as a unique pair of arc and chord of sector of a circle irrespective of arc radius. Two theorems viz. Equal arcs have equal central angles and equal chords have equal central angles are combined to establish a unique relationship between a pair of arc-chord and its composite of three identical pairs of arc-chord, thereby revealing a CYCLIC TRAPEZIUM, where the base defines the angle, and each equal edge defines each of the equal trisected parts of this angle. For a range of angles between 0o and 360o, this relationship is expressed as Lorna Graph, which becomes the practical tool for trisection of any angle, using the working backwards approach. This approach is extended to division of any angle into any number of equal parts.
Published in | American Journal of Applied Mathematics (Volume 3, Issue 4) |
DOI | 10.11648/j.ajam.20150304.11 |
Page(s) | 169-173 |
Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
Copyright |
Copyright © The Author(s), 2015. Published by Science Publishing Group |
Cyclic Trapezium, Working Backwards, Practical Tool
[1] | “Trisecting any angle” (2014, August 22) retrieved from http://math.stackexchange.com/questions/16641/trisect-unknown-angle-using-pencil-straight-edge-compass-prove-validity-of-technique |
[2] | Tapson, Frank, “Oxford Mathematics Study Dictionary”, p.46, Oxford University Press, 2006 |
[3] | Claphan, Christopher, “The concise Oxford Dictionary of Mathematics Oxford Reference”, p. 184 Oxford University Press, New York, 1990 |
[4] | Edited by Iyanaga, Shokichi; Kawada, Yukiyosi (translated by Mathematical Society of Japan, American Mathematical Society) “Encyclopedic Dictionary of Mathematics: Mathematical Society of Japan” p. 588 MIT press, Cambridge 1968 |
[5] | “Trisection of angles” (2014, September 6) retrieved from http://www.amsi.org.au/teacher_modules/Construction.html |
[6] | “Trisection” (2014, November 15) retrieved from http://www.cut-the-knot.org/Curriculum/Geometry/Vjecsner.shtml |
[7] | “Trisection of any angle” (2014, November 15) retrieved from http://archive.lib.msu.edu/crcmath/math/math/t/t383.htm |
[8] | “Trisecting an angle” (2014, November 17) retrieved from http://mathandmultimedia.com/2013/07/07/3-ancient-construction-problems/ |
[9] | Encyclopedia International, Grolier Incorporated, New York/Montreal/Mexico City/Sydney 1965,1964, 1963 p. 229 |
[10] | Encyclopedia Britannica Inc. 15th Edition, p.130. William Benton Publisher 1943-1973; Helen Hemingway Benton Publisher 1973-1974 Founded 1968, © 1978 |
[11] | Penney, David E “Perspectives in Mathematics”, p.18 The University of Georgia, W.A Benjamin Inc. Menlo Park, California ©1972 |
[12] | Robinson, Gilbert D.B, Collier’s Encyclopedia with Bibliographical Index Vol 10/24; p. 688Maxwell Macmillan International Publishing Group Canada 1992 |
[13] | Bennett, Albert B Jr, Nelson, l. Ted “Mathematics of Elementary Teacher: A Conceptual Approach” 5th Edition p.722. Mc Graw Hill Higher Education ISBN 0-07-23481-7 © 2001 |
[14] | New Encyclopedia Britannica, “Macropedia-Knowledge in Depth” Vol 19 Founded 1768 15th Edition, Inc. p.890 Jacob E. Safra, Chairman of the board Jorge Aguillar-Cuiz, President. London/New Delhi/Paris/Seoul/Sydney/Taipei/Tokyo © 2005 |
[15] | Gardner, Martin “Entertaining Science: Experiments with everyday”, illustrated by Anthony Ravielli (n.d, publisher not given) |
[16] | J.F.S., Chamber’s Encyclopedia New Revised Edition Vol 13 p.791Spain-Turing International Learning Systems Corporation Limited, London © 1966 Pergamon Press |
APA Style
Lorna A. Willis. (2015). Trisection of Any Angle and Consequentially the Division of Any Angle Into Any Number of Equal Parts. American Journal of Applied Mathematics, 3(4), 169-173. https://doi.org/10.11648/j.ajam.20150304.11
ACS Style
Lorna A. Willis. Trisection of Any Angle and Consequentially the Division of Any Angle Into Any Number of Equal Parts. Am. J. Appl. Math. 2015, 3(4), 169-173. doi: 10.11648/j.ajam.20150304.11
AMA Style
Lorna A. Willis. Trisection of Any Angle and Consequentially the Division of Any Angle Into Any Number of Equal Parts. Am J Appl Math. 2015;3(4):169-173. doi: 10.11648/j.ajam.20150304.11
@article{10.11648/j.ajam.20150304.11, author = {Lorna A. Willis}, title = {Trisection of Any Angle and Consequentially the Division of Any Angle Into Any Number of Equal Parts}, journal = {American Journal of Applied Mathematics}, volume = {3}, number = {4}, pages = {169-173}, doi = {10.11648/j.ajam.20150304.11}, url = {https://doi.org/10.11648/j.ajam.20150304.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150304.11}, abstract = {Historically, a contrived trisected line was used to trisect any other line, using the principle of projection. This is in essence about relationship and its accomplishment is about working backwards. Loosely speaking, any angle comprises two connecting lines. Attempts at trisecting any angle, which is dividing it into three equal parts, failed. In this paper any angle is defined as a unique pair of arc and chord of sector of a circle irrespective of arc radius. Two theorems viz. Equal arcs have equal central angles and equal chords have equal central angles are combined to establish a unique relationship between a pair of arc-chord and its composite of three identical pairs of arc-chord, thereby revealing a CYCLIC TRAPEZIUM, where the base defines the angle, and each equal edge defines each of the equal trisected parts of this angle. For a range of angles between 0o and 360o, this relationship is expressed as Lorna Graph, which becomes the practical tool for trisection of any angle, using the working backwards approach. This approach is extended to division of any angle into any number of equal parts.}, year = {2015} }
TY - JOUR T1 - Trisection of Any Angle and Consequentially the Division of Any Angle Into Any Number of Equal Parts AU - Lorna A. Willis Y1 - 2015/06/14 PY - 2015 N1 - https://doi.org/10.11648/j.ajam.20150304.11 DO - 10.11648/j.ajam.20150304.11 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 169 EP - 173 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20150304.11 AB - Historically, a contrived trisected line was used to trisect any other line, using the principle of projection. This is in essence about relationship and its accomplishment is about working backwards. Loosely speaking, any angle comprises two connecting lines. Attempts at trisecting any angle, which is dividing it into three equal parts, failed. In this paper any angle is defined as a unique pair of arc and chord of sector of a circle irrespective of arc radius. Two theorems viz. Equal arcs have equal central angles and equal chords have equal central angles are combined to establish a unique relationship between a pair of arc-chord and its composite of three identical pairs of arc-chord, thereby revealing a CYCLIC TRAPEZIUM, where the base defines the angle, and each equal edge defines each of the equal trisected parts of this angle. For a range of angles between 0o and 360o, this relationship is expressed as Lorna Graph, which becomes the practical tool for trisection of any angle, using the working backwards approach. This approach is extended to division of any angle into any number of equal parts. VL - 3 IS - 4 ER -