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This postulate does not specifically talk about parallel lines; [1] it is only a postulate related to parallelism. Euclid gave the definition of parallel lines in Book I, Definition 23 [2] just before the five postulates. [3] Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate.
The definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements. [7] Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate .
Two straight lines, meeting at a point, are not both parallel to a third line. This brief expression of Euclidean parallelism was adopted by Playfair in his textbook Elements of Geometry (1795) that was republished often. He wrote [8] Two straight lines which intersect one another cannot be both parallel to the same straight line.
Simply replacing the parallel postulate with the statement, "In a plane, given a point P and a line l not passing through P, all the lines through P meet l", does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry, [21] but this statement says that there are no parallel lines. This problem was ...
The most natural setting for Pascal's theorem is in a projective plane since any two lines meet and no exceptions need to be made for parallel lines. However, the theorem remains valid in the Euclidean plane, with the correct interpretation of what happens when some opposite sides of the hexagon are parallel.
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels.
The distance between two parallel lines in the plane is the minimum distance between any two points. Formula and proof. Because the lines are parallel, the ...
The de Longchamps point is the point of concurrence of several lines with the Euler line. Three lines, each formed by drawing an external equilateral triangle on one of the sides of a given triangle and connecting the new vertex to the original triangle's opposite vertex, are concurrent at a point called the first isogonal center.