- When
*B*is between*A*and*C*then,*A*,*B*and*C*are distinct points lying on a line and*B*is between*C*and*A*. - Given a pair of points
*A*and*B*there is a point*C*so that*B*is between*A*and*C*. - If
*B*lies between*A*and*C*then*A*does not lie between*B*and*C*. - Let
*A*,*B*and*C*be three points on a plane and*a*be a line on that does not contain any one of these points. If there is a point*D*on*a*that is between*A*and*B*then either*a*contains a point between*A*and*C*or*a*contains a point between*B*and*C*.

picture(6309,4845)(2209,-4531)
(5551,179)(0,0)[lb]*A*
(2446,-3646)(0,0)[lb]*B*
(6241,-4531)(0,0)[lb]*C*
(3976,-1561)(0,0)[lb]*D*
(2296,-1231)(0,0)[lb]*l*

The Line*l* contains a point on one of the other two sides

The first three axioms allow us to introduce the notion of a half-line
or ray. Given a pair of points The Line

In spite of the axioms of order being ignored for so many hundreds of years they are so important that one can entirely replace the axioms of incidence by giving an extended set of axioms of order. Think of it this way. If a straight line is to be the shortest path from a point to another then we must at least be able to say what are the points ``on the way'' or in-between.

The following theorems can be deduced from the axioms of Incidence and Order.

picture(8379,7452)(949,-6688)
(4366,-3556)(0,0)[lb]*A*
(4201,-6571)(0,0)[lb]*A'*
(3076,-4066)(0,0)[lb]*B'*
(3526,-2716)(0,0)[lb]*B*
(5581,-3136)(0,0)[lb]*C*
(5716,-3826)(0,0)[lb]*C'*
(6331,-2326)(0,0)[lb]*B''*
(2041,-826)(0,0)[lb]*C''*
(8251,-3076)(0,0)[lb]*A''*
(4936,629)(0,0)[lb]*O*

The triangles*ABC* and *A'B'C'* lie in different planes.

In the remaining cases we examine all the possibilities for
between-ness for the triples (The triangles

*A*does not lie between*A'*and*O*,*B*does not lie between*B'*and*O*,*C*does not lie between*C'*and*O*.*A'*lies between*A*and*O*,*B'*does not lie between*B*and*O*,*C'*does not lie between*C*and*O*.

picture(9549,6694)(139,-7790)
(601,-2611)(0,0)[lb]*A''*
(1201,-7336)(0,0)[lb]*A*
(2551,-5236)(0,0)[lb]*B*
(5026,-7261)(0,0)[lb]*C*
(7501,-4186)(0,0)[lb]*O*
(4351,-6211)(0,0)[lb]*A'*
(3751,-5236)(0,0)[lb]*B'*
(5776,-6436)(0,0)[lb]*C'*
(5251,-3211)(0,0)[lb]*A'''*
(4501,-3436)(0,0)[lb]*B'''*
(3526,-3811)(0,0)[lb]*C''*
(5926,-3736)(0,0)[lb]*C'''*
(9151,-6211)(0,0)[lb]*B''*
(6076,-2011)(0,0)[lb]*O''*
(5476,-1261)(0,0)[lb]*O'*

Lifting*A'B'C'* in the first case.

Examining the first case, let Lifting

picture(5850,7924)(1351,-8461)
(4651,-8461)(0,0)[lb]*B*
(7201,-5761)(0,0)[lb]*C'*
(5326,-3661)(0,0)[lb]*B'*
(5476,-4786)(0,0)[lb]*O*
(3526,-6436)(0,0)[lb]*A'*
(1426,-961)(0,0)[lb]*A'''*
(3226,-5011)(0,0)[lb]*O'*
(4276,-4036)(0,0)[lb]*C*
(4051,-1936)(0,0)[lb]*A''*
(2551,-4786)(0,0)[lb]*C'''*
(2776,-6436)(0,0)[lb]*B''*
(1651,-5761)(0,0)[lb]*B'''*
(1351,-5311)(0,0)[lb]*O''*
(1531,-7636)(0,0)[lb]*A*
(2341,-7786)(0,0)[lb]*C''*

Lifting*A'B'C'* in the second case.

In the second case we choose a point Lifting

In both these cases the line joining *A'''* and *B'''* meets the plane
within the intersection of the plane determined by *A'*, *B'*
and *O'* and the plane ; this is the line joining *A'* and
*B'*. Similarly, the intersection of the line joining *A'''* and
*B'''* with lies within the intersection of the plane
determined by *A*, *B* and *O''* with the plane ; this is the
line joining *A* and *B*. In other words, the line joining *A'''* and
*B'''* contains the point *C''*. We prove the other containments
cyclically.

picture(5454,6039)(1339,-7138)
(1396,-3166)(0,0)[lb]*l*
(2191,-4951)(0,0)[lb]*m*
(2911,-5056)(0,0)[lb]*A''*
(2866,-2506)(0,0)[lb]*B*
(3526,-3691)(0,0)[lb]*B'*
(4651,-5521)(0,0)[lb]*O*
(2281,-6571)(0,0)[lb]*C'*
(3421,-6301)(0,0)[lb]*C*
(4786,-1741)(0,0)[lb]*A*
(4606,-3046)(0,0)[lb]*A'*
(4081,-4471)(0,0)[lb]*B''*

The line joining*A''* and *B''* passes through the intersection of
*l* and *m*.

Another aspect of Desargues' theorem is that it's proof makes use of
the non-planar axioms of incidence. We shall see that
The line joining