A disk model for euclidean geometry

We have already met i-lines which meet C twice, but not orthogonally - hypercircles.
We showed that they could be defined as loci. In detail, if P is a point and L is a
hyperbolic line not through P, we defined the hypercircle C(P,L) as the set of points
on the same side of L as P, and equally distant (in the hyperbolic metric) from L.
If P is on L, we simply say C(P,L) = L. If we take L to be a diameter of C, then we
get the e-lines with the ends of the diameter as boundary points.

Of course, this still uses the idea of a diameter, but this is simply a hyperbolic line
through O. Now we have a purely hyperbolic description :
an e-line is a locus C(P,L), where P is a point in D, and L a hyperbolic line through O.

We shall refer to L as the hyperbolic line associated with the e-line.

e-line theorem
  1. Two distinct e-lines either have the same associated hyperbolic line
    (and do not meet in D), or meet exactly once in D.
  2. Given two distinct points A,B, there is exactly one e-line through A and B.
  3. Given the e-line E and the point A, there is exactly one e-line through A
    parallel to E.


Since our grasp on e-lines as loci is somewhat tenuous, the proofs of (1) and (2) are
rather tricky. On the other hand (3) is easy. Let L be the hyperbolic line associated
with E. Any e-line parallel to E must also be associated with L, by (1). The only choice
is C(A,L) if it is to pass through A.

To verify that the geometry of e-lines is indeed euclidean, we use some algebra.

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