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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 postulate was long considered to be obvious or inevitable, but proofs were elusive.
Line art drawing of parallel lines and curves. In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three ...
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 "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. A proof from Euclid's Elements that, given a line segment, one may construct an equilateral triangle that includes the segment as one of its sides: an equilateral triangle ΑΒΓ is made by drawing circles Δ and ...
Pappus' law: if the red lines are parallel and the blue lines are parallel, then the dotted black lines must be parallel. As affine geometry deals with parallel lines, one of the properties of parallels noted by Pappus of Alexandria has been taken as a premise: [9] [10] Suppose A, B, C are on one line and A', B', C' on another.
The theorems of absolute geometry hold in hyperbolic geometry, which is a non-Euclidean geometry, as well as in Euclidean geometry. [9] Absolute geometry is inconsistent with elliptic geometry: in that theory, there are no parallel lines at all, but it is a theorem of absolute geometry that parallel lines do exist. However, it is possible to ...
Poincaré disc: The pink line is ultraparallel to the blue line and the green lines are limiting parallel to the blue line. In hyperbolic geometry, two lines are said to be ultraparallel if they do not intersect and are not limiting parallel. The ultraparallel theorem states that every pair of (distinct) ultraparallel lines has a unique common ...
Antecedent of Playfair's axiom: a line and a point not on the line Consequent of Playfair's axiom: a second line, parallel to the first, passing through the point. In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):
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