Sec. 2.1 Lines, segments and half-lines 23
2.1.3 Reciprocalorders
IfA≤lBwe also writeB≥lAand read this as ‘Bsucceeds or coincides withAon
l’. Then≥lis also a total order onl,i.e.≥lsatisfies A 2 (i), (ii), (iii) and (iv), as can
readily be checked as follows.
First, on interchangingAandAin A 2 (i), we haveA≥lAfor allA∈l. Secondly,
suppose thatA≥lBandB≥lC;thenC≤lBandB≤lA,sobyA 2 (ii)C≤lA; hence
A≥lC. Thirdly, suppose thatA≥lBandB≥lA;thenB≤lAandA≤lBso by A 2 (iii)
A=B. Finally, letA,Bbe any points onl;byA 2 (iv), eitherA≤lBorB≤lAand so
eitherB≥lAorA≥lB.
We say that≥lisreciprocalto≤l. If now we start with≥land let (^) lbe its
reciprocal we haveA (^) lBifB≥lA;thenwehaveA (^) lBif and only ifA≤lB. Thus
(^) lcoincides with≤l, and so the reciprocal of≥lis≤l.
The upshot of this is that≤land≥lare a pair of total orders onl, each the recipro-
cal of the other. There is no natural way of singling out one of≤l,≥lover the other,
and the notation is equally interchangeable as we could have started with≥l.Having
this pair is a nuisance but it is unavoidable, and we try to minimise the nuisance as
follows. Given distinct pointsAandB,letl=AB. Then exactly one ofA≤lB,A≥lB
holds; for by A 2 (iv) eitherA≤lBorA≥lB, and by A 2 (iii) both cannot hold as that
would imply thatA=B. Thus we can choose the natural order onlin whichApre-
cedesB,bytaking≤lwhenA≤lB, and by taking≥lwhenA≥lB; we will use the
notation≤lfor this natural order.
Let A and B be distinct points inΠ,letl=AB and A≤lB. Let C be a point of l,
distinct from A and B. Then exactly one of
(a)C≤lA≤lB,(b)A≤lC≤lB,(c)A≤lB≤lC,
holds.
Proof.IfC≤lAthen clearly (a) holds. IfC≤lAis false, then by A 2 (iv)A≤lC;
by A 2 (iv) we have moreover eitherC≤lBorB≤lC, and these yield, respectively,
(b) and (c). Thus at least one of (a), (b), (c) holds.
On the other hand, if (a) and (b) hold, we haveA=Cby A 2 (iii) and this contradicts
our assumptions. Similarly if (b) and (c) hold we would haveB=C. Finally if (a)
and (c) hold, from (a) we haveC≤lBby A 2 (ii) and thenB=C.
2.1.4 Segments................................
Definition. For any pointsA,B∈Π,wedefinethesegments[A,B]and[B,A]as
follows. Letlbe a line such thatA,B∈land≤l,≥la pair of reciprocal natural
orders onl.Thenif
A≤lB so that B≥lA, (2.1.1)
we define
[A,B]={P∈l:A≤lP≤lB}={P∈l:A≤lPandP≤lB},
[B,A]={P∈l:B≥lP≥lA},