Incidence Propositions

    Hilbert partitioned his axioms for Euclidean geometry into five groups (connection, order, congruence, parallels, continuity) and his axioms of connection (or incidence) will shed some light on his undefined terms. Hilbert's axioms went a long way towards helping people understand that the axiomatic approach to mathematics was a viable and much needed mathematical enterprise in its own right. Many axiomatic systems have been created and even recreated for subjects that are widely understood in the mathematical community. Hilbert's axioms are close in spirit to Euclid's postulates because they do not associate real numbers with segments; i.e. there is no way to measure a line segment. This section details Hilbert's incidence axioms and illustrates some basic propositions that can be proven directly from these axioms.

Proposition (Point Uniqueness) If and are distinct lines that are not parallel, then and have a unique point in common.

    Proof. By the Parallel Lines Definition, and have at least one point in common. Suppose that and have two distinct points and in common. By the Line Uniqueness Axiom, and must be the same line, which is a contradiction to the hypothesis. Therefore, and must be the same point.

Proposition (Non-Concurrent Lines) There exist three distinct lines that are not concurrent.

    Proof. By the Non-Collinear Points Axiom, there exists three distinct non-collinear points and and no line is incident with all three. By the Line Uniqueness Axiom, there exist unique lines and which are not concurrent.  

Proposition (Point Not On Line) For every line there is at least one point not lying on it.

    Proof.  Suppose there is a line that contains all points. By the Non-Collinear Points Axiom there are three points on this line that no line is incident with, which is a contradiction. Therefore, there is no line with all points on it.  

Proposition (Line Missing Point) For every point there is at least one line not passing through it.

    Proof. Given any point assume that every line goes through By the Non-Concurrent Lines Proposition there exist three distinct lines that are not concurrent, which is a contradiction. Therefore, every line can not go through   

Proposition (Lines Through Point) For every point there exist at least two lines through

    Proof. Suppose that there is a point that has no line through By the Non-Collinear Axiom there exists three non-collinear points say, and By the Law Of The Excluded Middle either is one of these three points or not. Suppose is one of these points, say Then by the Line Uniqueness Axiom there exists a line though and If is not one of these three points, then by the Line Uniqueness Axiom there exists a line through and All cases show that there is at least one line through
    Suppose there is a point that has only one line passing through it. By  the Point Not On Line Proposition, there exist a point not lying on this line, say . So there are two distinct points and , and by the Line Uniqueness Axiom, there is a unique line through these two points. Since we are assuming there is only one line through and both lines pass through the lines and must be the same. This is a contradiction because is not on and is on Therefore, must have at least two lines through it.  

• print this page
• pages linking here
• related pages

Cite this as:
Incidence Propositions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/incidence-propositions.html