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The Elements contains the proof of an equivalent statement (Book I, Proposition 27): If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. As De Morgan [23] pointed out, this is logically equivalent to (Book I, Proposition 16).
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.
Formula and proof. Because the lines are parallel, the perpendicular distance between them is a constant, so it does not matter which point is chosen to measure the ...
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 ...
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 perpendicular (a hyperbolic line which is perpendicular to both lines).
Proposition 31 is the construction of a parallel line to a given line through a point not on the given line. [6] As the proof only requires the use of Proposition 27 (the Alternate Interior Angle Theorem), it is a valid construction in absolute geometry. More precisely, given any line l and any point P not on l, there is at least one line ...
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 ...
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):