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30. If the four sides of a gauche quadrilateral touch a sphere, the points of contact are concyclic.
NOTES.
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NOTE A.
MODERN THEORY OF PARALLEL LINES.
In every plane there is one special line called the line at infinity. The point where any other line in the plane cuts the line at infinity is called the point at infinity in that line. All other points in the line are called finite points. Two lines in the plane which meet the line at infinity in the same point are said to have the same direction, and two lines which meet it in different points to have different directions. Two lines which have the same direction cannot meet in any finite point [I. Axiom x.], and are parallel. Two lines which have different directions must intersect in some finite point, since, if produced, they meet the line at infinity in different points. This is a fundamental conception in Geometry, it is self-evident, and may be assumed as an Axiom (see Observations on the Axioms, Book I.). Hence we may infer the following general proposition:—“Any two lines in the same plane must meet in some point in that plane; that is—(1) at infinity, when the lines have the same direction; (2) in some finite point, when they have different directions.”—See Poncelet, Propriétés Projectives, page 52.
NOTE B.