Ultraparallel theorem: Difference between revisions

Revision as of 03:44, 21 April 2006

Revision as of 08:28, 4 May 2006

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chords are diameters, they intersect. If only one of the chords is a diameter, the

other chord projects orthogonally down to a section of the first chord contained in its interior, and a line from the pole orthogonal to the diameter intersects both the diameter and the chord. If both lines are not diameters, the we may extend the tangents drawn from each pole to produce a quadrilateral with the unit circle inscribed within it. The poles are opposite vertices of this quadrilateral,

and the chords are lines drawn between adjacent sides of the vertex, accrossacross opposite corners. Since the quadrilateral is convex, the line between the poles intersects both of the chords drawn across the corners, and the segment of the line between the chords defines the required chord perpendicular to the two other chords.

==Proof in the Poincaré half-plane model==