We have already met ilines 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 elines 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 We shall refer to L as the hyperbolic line associated with the eline.


eline theorem
Since our grasp on elines as loci is somewhat tenuous, the proofs of (1) and (2) are To verify that the geometry of elines is indeed euclidean, we use some algebra. 
