Geometry/Neutral Geometry/Axioms of Betweenness
< Geometry | Neutral Geometry
Axioms of Betweenness[edit | edit source]
Betweenness Axiom 1[edit | edit source]
If A*B*C, then A,B, and C are three distinct points all lying on the same line, and C*B*A.
Explanation[edit | edit source]
Betweenness Axiom 2[edit | edit source]
Given any two distinct points B and D, there exist points A,C, and E lying on line BD (needs format with LaTex) such that A*B*D, B*C*D, and B*D*E.
Explanation[edit | edit source]
suppose that in a certain metric geometry the following distance relationship hold: AB= 2 AD=BD=CD=3 BC=4 AC=6
Betweenness Axiom 3[edit | edit source]
If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two.
Explanation[edit | edit source]
Betweenness Axiom 4[edit | edit source]
For every line l and for any three points A, B, and C not lying on l:
- (i) If A and B are on the same side of l and if B and C are on the same side of l, then A and C are on the same side of l.
- (ii) If A and B are on opposite sides of l and if B and C are on opposite sides of l, then A and C are on the same side of l.
Corollary[edit | edit source]
- (iii) If A and B are on opposite sides of l and if B and C are on the same side of l, then A and C are on opposite sides of l.