Editing Euclidean geometry (section)

===Parallel postulate===
{{main|Parallel postulate}}
To the ancients, the parallel postulate seemed less obvious than the others. They aspired to create a system of absolutely certain propositions, and to them, it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible since one can construct consistent systems of geometry (obeying the other axioms) in which the parallel postulate is true, and others in which it is false.<ref>{{Citation|title=History of the Parallel Postulate|journal=The American Mathematical Monthly|volume=27|issue=1|pages=16–23|date=Jan 1920|author=Florence P. Lewis|doi=10.2307/2973238|publisher=The American Mathematical Monthly, Vol. 27, No. 1|postscript=.|jstor=2973238}}</ref> Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the ''Elements'': his first 28 propositions are those that can be proved without it.

Many alternative axioms can be formulated which are [[logical equivalence|logically equivalent]] to the parallel postulate (in the context of the other axioms). For example, [[Playfair's axiom]] states:

:In a [[Plane (geometry)|plane]], through a point not on a given straight line, at most one line can be drawn that never meets the given line.

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.

[[File:euclid-proof.svg|thumb|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 Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle.]]