The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Math Topics » geometry.puzzles

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Trisection
Replies: 19   Last Post: Feb 11, 2012 9:59 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 35
Registered: 12/4/04
Trisection (and pentasection and septasection etc)
Posted: Jul 3, 2002 10:52 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Very interesting Mark. I would like to share my schooltime method
with you. Unlike yours, it is very easy to prove why my method is
wrong but, nevertheless, it is quite accurate.
Start with any angle (<45 degrees) and label the vertex A. Step off
three equal distances on one leg and label them B, C and D.
With A as centre draw three concentric arcs (radii AB, AC and AD) to
cut the to cut the other leg in E, F and G resp.
Bisect angle A to cut the three arcs BE, CF and DG in H, J and K

With compass step off distance CJ on arc DG; it trisects the arc &
thereby trisects angle A. It is easy to see that arc CJ is actually
the length that constitutes a third of arc CJ but the difference is
virtually insignificant if angle A < 45 degrees. Therefore, if an
angle is more than, say 60 degrees, bisect it and do the operation on
half the angle and double afterwards.

Arc BE is also a third of arc DG but because of the greater curvature
("fatter" curve) distance BE is significantly shorter than arc BE and
will fit slightly less than three times into curve DG. Arc CF is 2/3
arc DG (^A*AC = 2/3 of ^A*AD because AC is 2 units and AD is three
units, ^A in radians of course).

In the same way any arc can be subdivided into 5, 7 or whatever by
just drawing the appropriate number of concentric arcs.

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.