If A, B are two points on a line a, and if A′ is a point upon the same or another line a′, then, upon a given side of A′ on the straight line a′, we can always find a point B′ so that the segment AB is congruent to the segment A′ B′.Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC. Pasch's Axiom: Let A, B, C be three points not lying in the same line and let a be a line lying in the plane ABC and not passing through any of the points A, B, C.Of any three points situated on a line, there is no more than one which lies between the other two.If A and C are two points, then there exists at least one point B on the line AC such that C lies between A and B.If a point B lies between points A and C, B is also between C and A, and there exists a line containing the distinct points A, B, C.There exist at least four points not lying in a plane.If two planes α, β have a point A in common, then they have at least a second point B in common.In this case we say: "The line a lies in the plane α", etc. If two points A, B of a line a lie in a plane α, then every point of a lies in α.For every three points A, B, C which do not lie in the same line, there exists no more than one plane that contains them all.We employ also the expressions: " A, B, C lie in α" " A, B, C are points of α", etc. For every plane there exists a point which lies on it. For every three points A, B, C not situated on the same line there exists a plane α that contains all of them.There exist at least three points that do not lie on the same line. There exist at least two points on a line. ![]() For every two points there exists no more than one line that contains them both consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.If A lies upon a and at the same time upon another line b, we make use also of the expression: "The lines a and b have the point A in common", etc. Instead of "contains", we may also employ other forms of expression for example, we may say " A lies upon a", " A is a point of a", " a goes through A and through B", " a joins A to B", etc. For every two points A and B there exists a line a that contains them both.All points, straight lines, and planes in the following axioms are distinct unless otherwise stated. Line segments, angles, and triangles may each be defined in terms of points and straight lines, using the relations of betweenness and containment. Congruence, two binary relations, one linking line segments and one linking angles, each denoted by an infix ≅. ![]() ![]() Lies on (Containment), three binary relations, one linking points and straight lines, one linking points and planes, and one linking straight lines and planes.Betweenness, a ternary relation linking points.Hilbert's axiom system is constructed with six primitive notions: three primitive terms: Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry. ![]() Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr.
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