From: 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 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 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)