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In hyperbolic geometry, the ultraparallel theorem states that every pair of ultraparallel lines (lines that are not intersecting and not limiting parallel) has a unique common perpendicular hyperbolic line. ==Hilberts construction== Let r and s be two non-intersecting lines. From any two points A and C on s draw AB and CB' perpendicular to r. (B and B' on r) If it happens that AB = CB' the desired common perpendicular joins the midpoints AC and BB'(by the symmetry of the isocleses birectangle ACB'B ). . If not, suppose AB < CB'. Take A' on CB' so that A'B' = AB. Trough A' draw a line s', making the same angle with A'B' that s makes with AB. Then s meets s' in an ordinary point D. Take a point D' on ray AC so that AD' = A'D. Then the perpendicular bisector of DD' is also perpendicular to r. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ultraparallel theorem」の詳細全文を読む スポンサード リンク
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