🚀 go-pugleaf

The solution to the compass construction problem can be found in
"What is Mathematics" by Richard Courant, a real classic, equally
enjoyable to the recreational mathematician or to the professional.
I don't have it now so I can't give you the publisher or year.

Being unable to draw a diagram I can't describe the construction
adequately but I'll try to cover the main points. Remember that
you are using only a compass, so whenever you "mark a point" it
is done by intersecting one arc with another arc. The straight
line joining the two points in the "diagram" is for clarity only and
is not actually drawn for you in the problem.

	.________.________.
	A        B        C

You start with two points A and B, now obtain a point C on AB extended,
so that AB=BC. Draw a circle about B with radius BA and C will be the
point diametrically opposite to A on this circle. Find this point by 
marking off points along the circumference, starting at A as centre, 
with radius BA. Call the first point A', then mark the next point 
using A' as the centre, call this A" and the next point will be C 
(like the construction of a regular hexagon inscribed in a circle).

Once you have found C, you have to find the inverse point w.r.to C on the
line AB. The inverse point is defined as a point D on AB, such that
AD/AB = AC/BC. Since in this case AC=2.AB, you will have AD=AB/2.

The construction is as follows, proof by similar triangles not included:

Draw a circle centred at A, radius AB. Use C as centre and mark points
P and P' on the circumference of this circle, by drawing an arc with 
radius CA. Now draw arcs centred at P and P' respectively, with radius
PB. These arcs will intersect at 2 points, one is B itself, the
^^
other is the point D which is the bisection point you are looking for.
For proof, consider the similar triangles APC and CPD, and remember the
definition of the inverse point.

		* End of construction *

Now that you know how to get the bisection point, it should be simple
for you to obtain the 1/n point for any integer n, using compasses
alone.

(Where I come from they spell "center" as "centre". "Spelling flames
are a hobgoblin of small minds" - misquoted from Emerson).


Vivek Rau.
vivekr@osc.com