Axiom  Manipulative  Discussion 
II,1 
Click on the blue points and drag them the change the figure.
Note that point B is between point A and C, and the B is between points C and A and B is distinct from A and C.
 Manipulative 1  Hilbert Axiom ii1 Created with GeoGebra. 

"If A, B, and C are points of a
straight line and B lies between A and C, then
B lies also between C and A."
This tells us that if point B is between A and C,
then it is also between points C and A.

II,2 
Click on the blue points and drag them to change the figure.
There always exists a point B on line AC such that C lies between A and B.
 Manipulative 2  Hilbert Axiom ii2 Created with GeoGebra. 

"If A and C are two points of a straight line,
then there exists at least one point B lying between A and
C and at least one point D so situated that C lies
between A and D."
This axiom establishes that between any two points on a line, there exists
another point. A consequence of this axiom is that an
infinite
number of points lie between any two points on a line.

II,3 
Click on the blue points and drag them to change the figure.
Can you arrange the points on the line so that two of them are between other points?
 Manipulative 3  Hilbert Axiom ii3 Created with GeoGebra. 

"Of any three points situated on a straight line, there is always one
and only one that lies between the other two."
This says that if point B lies between points A and
C then point A can not lie between points
B and C.

II,4 
Click on the blue points and drag them to change the figure
This axiom establishes that it is possible to order points on a line.
 Manipulative 4  Hilbert Axiom ii4 Created with GeoGebra. 

"Any four points A, B, C, D
of a straight line can always be arranged so that B shall lie between
A and C and also between A and D, and,
furthermore, that C shall lie between A and D
and also between B and D."
This axiom establishes that points on a line can always be put in order.

II,5 
Click on the blue and red points and drag them to change the figure.
Is it possible for the red line to cross one of the sides of the triangle without crossing either of the other sides?
 Manipulative 5  Hilbert Axiom ii5 Created with GeoGebra. 

"Let A, B, C be points not lying
in the same straight line and let a be a straight line lying in the
plane ABC and not passing through any of the points A,
B, and C.
Then, if a straight line passes through a point of segment AB, it will
always pass through either a point of the segment BC or a point of the segment
AC."
This axiom establishes that a straight line that intersects one side of a triangle, must
also intersect a second side of the same triangle.
