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In geometry, Pasch's axiom is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882. ==Statement== The axiom states that,〔This is taken from the Unger translation of the 10th edition of Hilbert's ''Foundations of Geometry'' and is numbered II.4.〕 :Let A, B, C be three points that do not lie on a line and let ''a'' be a line in the plane ABC which does not meet any of the points A, B,C. If the line ''a'' passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of segment BC. The fact that both segments AC and BC are not intersected by the line ''a'' is proved in Supplement I,1, which was written by P. Bernays.〔, the Unger translation.〕 A more modern version of this axiom is as follows: :In the plane, if a line intersects one side of a triangle ''internally'' then it intersects precisely one other side ''internally'' and the third side ''externally'', if it does not pass through a vertex of the triangle. (In case the third side is parallel to our line, we count an "intersection at infinity" as external.) A more informal version of the axiom is often seen: :If a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side. 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「Pasch's axiom」の詳細全文を読む スポンサード リンク
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