🚀 go-pugleaf

In article <33mbhc$gug@nermal.cs.uoguelph.ca>,
Douglas Harder <dharder@sgi2.mathstat.uoguelph.ca> wrote:
:Noah (deh@access1.digex.net) wrote:
:: 1) What are the three constructions in Euclidian geometry that are 
:: impossible to achieve with only a compass and a straightedge?
:
:One is the trisection of an angle.
:
:I believe another is finding a focus of any conic which is
:not a circle.
:
:I don't know the third.
:
Hmm. I think they were trisection of the angle, doubling a cube, and
squaring a circle. The one you mention may be equivalent to these; I can't
say offhand.

:He're another point about Euclidean geometry:
:
:Postulate 2 reads that given any line segment, you can construct an
:equalateral triangle with that line segment as a side.
:
:Lets prove it using Euclids axioms.
:
:We can draw circles around each end of the line segment
:with radius equal to the length of the line segment.
:
:Let the intersection of the two circles be at a point C. ****
:
:Connect the ends of the line segment with the point C.
:
:Thus, we have constructed an equalateral triangle.
:
:
:This is fine, except for ****.  Nowhere in Euclid's axioms
:does he mention that two circles must intersect at a point --
:ie, Euclid does not guarentee us in his axioms that the
:point C exists.  We can see it on paper, but we can't 
:prove it without refering to a drawing.
:
The reason for this is because two circles indeed don't intersect each 
other if the sum of their radii is less than the distance between their
centers. Of course, this is not the case here. That such a point (OK, two
such points) exists follows almost trivially from Pythagorian theorem.
(Draw perpendicular bisector, etc.). Then drawing the circles *must*
yield two intersections because you are taking the intersection of two
loci which, as you just proved, consists of exactly two points. There may
be a way to prove it without using the fifth axiom (the infamous one about
a unique parallel line through a point, which is required by Pythagorian
Theorem), but I can't think of one off the top of my head....
-- 
Shimpei Yamashita, Stanford University           shimpei@leland.stanford.edu