From: AOUM <AOUM.>
Newsgroups: sci.math
Subject: Re: Problems from Greek Antiquity
Date: 17 Feb 1996 00:12:29 GMT

kochman@tang.ccr-p.ida.org (Fred Kochman) wrote:
>In article <4fmk6t$93r@digital.netvoyage.net> mjcohen@digital.netvoyage.net writes:
>
>   In article <4eio7b$qpp@news.ox.ac.uk>,
>   Gerald McLarnon <mclarnon@maths.ox.ac.uk> wrote:
>   >Does anyone know what the definitive treatment is on the three problems
>   >that caused the ancient Greeks so much trouble, namely
>   >
>   >1. Squaring the circle
>   >2. Doubling the cube
>   >3. Trisecting the angle
>   >
>   >using straight edge and compass alone (although I know there is some
>   >debate as to whether or not the Greeks placed such stringent requirements
>   >on the solutions to the problems). I have a vague notion about a
>   >Russian paper c1880 which proves (1) can't be done, but would be interested
>   >if anyone had definite dates and authors for proofs (I do not
>   >need to know actual references).
>   >
>   >The FAQ on trisecting the angle provides no reference (at least the version
>   >that I have).
>   >
>   >Thanks in advance.
>   >
>   >Gerald McLarnon
(snip snip ...)

Apollonius from Perga give us the following beutifull construction to 3.  :
 
               *
              *
             A*********************************o***************M
            *| 
           * |
          *  |
         *   |
        *    |
       *     |
      *      o
     *       |
    O********U********************************************M

* Given the Angel AOM draw a parallel AM to OM (A is arbitrary).
* Draw the perpendicular AU on OM.
* Now, take your compass opened twice the distance AO and let it slide
  on your ruler.
  Letting the ruler pass through the point O, 
  pass the distance 2*AO (opening of the ruler) in between the lines
  AU and AM.
  Your compass now points to the points o and o.
* The Angel AOUM is now trisected by the Angel ooOUM. 

==============================================================================

From: Roman Karawatzki <Roman.Karawatzki@univie.ac.at>
Newsgroups: sci.math
Subject: Re: Trisecting the Angle !
Date: 18 Feb 1996 03:22:16 GMT

AOUM (whatever or whoever he is) wrote :

> Apollonius from Perga give us the following beutifull construction to 3.  :
> 
>               *
>              *
>             A*********************************o***************M
>            *| 
>           * |
>          *  |
>         *   |
>        *    |
>       *     |
>      *      o
>     *       |
>    O********U********************************************M
>
> * Given the Angel AOM draw a parallel AM to OM (A is arbitrary).
> * Draw the perpendicular AU on OM.
> * Now, take your compass opened twice the distance AO and let it slide
>   on your ruler.
>   Letting the ruler pass through the point O, 
>   pass the distance 2*AO (opening of the ruler) in between the lines
>   AU and AM.
>   Your compass now points to the points o and o.
> * The Angel AOUM is now trisected by the Angel ooOUM. 

Yes, that's wonderful indeed !
There is even a simple geometric proof of that :

               *
              *
             A*********************************o***************M
            *|----                         ----| 
           * |     ----               ----     |
          *  |           ---      ---          |
         *   |               --u--             |
        *    |         ----          ----      |
       *     |   -----                     ----|
      *     -o---------------------------------m
     *  ---- |
    O---*****U********************************************M


Drawing the rectangular AomoA.
By construction, the distance OA equals the distances Au and ou.
The triangles AOu and Aou are therefore equilateral. 

Now, the angles (Angels :-) yes, funny) 

  <AOo = <AuO      (AOu is equiangular !)
       = 2 * <uAM  (yes, that's geometry !) 
       = 2 * <uoA  (the o between A and M)
       = 2 * <uOM  (AoM and OUM are parallel)

That's all folks, and it's true :-)