Axiom:Euclid's Fifth Postulate

's Statement
There are many equivalent ways to state this postulate. See below for a selection of them.

Other equivalent statements
Many further attempts have been made to formulate equivalent definitions of this axiom, often with a view to finding a proof which relies on the other four axioms. (Such attempts have been universally doomed to failure.) Here are a few examples, in approximate chronological order:


 * If a straight line intersects one of two parallels, it will intersect the other also.

Straight lines parallel to the same straight line are parallel to one another.

and

 * There exist straight lines everywhere equidistant from one another.

(This can be compared with ' tacit assumption that "Parallels remain, throughout their length, at a finite distance from one another.")


 * There exists a triangle in which the sum of the three angles is equal to two right angles.

, ,

 * Given any figure, there exists a figure similar to it of any size we please.

points out that it is necessary only to postulate that:
 * There exist two unequal triangles with equal angles.

(again)

 * Through any point within an angle less than two-thirds of a right angle a straight line can always be drawn which meets both sides of the angle.


 * Every straight line through a point within an angle must meet one of the sides of the angle.

and

 * Given any three points not in a straight line, there exists a circle passing through them.


 * If I could prove that a rectilineal triangle is possible the content of which is greater than any given area, I am in a position to prove perfectly rigorously the whole of geometry.


 * There exists no triangle in which every angle is as small as we please.


 * If in a quadrilateral three angles are right angles, the fourth angle is a right angle also. (1741)


 * If two straight lines are parallel, they are figures opposite to (or the reflex of) one another with respect to the middle points of all their transversal segments.


 * Two parallel straight lines intercept, on every transversal which passes through the middle point of a segment included within them, another segment the middle point of which is the middle point of the first.