26 Basic shapes of geometry Ch. 2
2.2 Open and closed half-planes
2.2.1 Convexsets
Definition. A setEis said to beconvexif for everyP,Q∈E,[P,Q]⊂Eholds.
NOTE. By 2.1.4(iv) every segment is a convex set; by 2.1.5(iii) so is every half-
line. In preparation for the next subsection, we note that by A 1 , for each linel∈Λ
we haveΠ\l=0./
2.2.2 Open half-planes
Primitive Term. Corresponding to each linel∈Λ, there is a pair {G 1 ,G 2 } of non-
empty sets calledopen half-planeswithcommon edgel.
AXIOM A 3 .Open half-planesG 1 ,G 2 with common edge l have the
properties:-
(i)Π\l=G 1 ∪G 2 ;(ii)G 1 andG 2 are both convex sets;(iii)if P∈G 1 and Q∈G 2 ,then[P,Q]∩l=0./ |We note the following immediately.
Open half-planes {G 1 ,G 2 } with common edge l have the properties:-
(i)l∩G 1 = 0 /,l∩G 2 =0./(ii)G 1 ∩G 2 =0./(iii)If P∈G 1 and[P,Q]∩l= 0 /where Q∈l, then Q∈G 2.(iv)Each line l determines a unique pair of open half-planes.Proof.
(i) By A 3 (i),l∩(G 1 ∪G 2 )=0andas/ G 1 ⊂G 1 ∪G 2 it follows thatl∩G 1 =0. The/
other assertion is proved similarly.
(ii) IfG 1 ∩G 2 =0, there is some point/ Rin bothG 1 andG 2 .ByA 3 (iii) withP=
R,Q=R,wehavethat[R,R]∩l=0. But/ Ris the only point in[R,R]soR∈l.This
contradicts the fact thatl∩G 1 =0./
(iii) For otherwise by A 3 (i),Q∈G 1 andthenbyA 3 (ii)[P,Q]⊂G 1 .Asl∩G 1 =0,/
it follows that[P,Q]∩l=0 which contradicts the assumptions./
(iv) Suppose that
Π\l=G 1 ∪G 2 =G 1 ′∪G 2 ′,
where{G 1 ,G 2 }and{G 1 ′,G 2 ′}are both sets of open half-planes with common edgel.
Then
G 1 ⊂G 1 ∪G 2 =G 1 ′∪G 2 ′
so at one of the following holds