Geometry with Trigonometry

24 Basic shapes of geometry Ch. 2

while if
B≤lA so that A≥lB, (2.1.2)

we define

[B,A]={P∈l:B≤lP≤lA},

[A,B]={P∈l:A≥lP≥lB}.

We should use a more complete notation such as[A,B]≤l,≥l,[B,A]≤l,≥l, but make do
with the less precise one. Note that (2.1.2) comes from (2.1.1) on interchangingA
andB, or on interchanging≤land≥l.
WhenA=B,by A 1 l=AB;by
A 2 (iv) at least one of (2.1.1) and
(2.1.2) holds, and by A 2 (iii) only
one of (2.1.1) and (2.1.2) holds.

WhenA=B,lcan be any line
throughA, and we find that{P∈
l:A≤lP≤lA}={A},{P∈
l:A≥lP≥lA}={A},forthe
singleton {A}. To see this we
note thatA≤lA≤lAby A 2 (i),
while ifA≤lP≤lAthenP=A
by A 2 (iii). The same argument
holds for≥l. Thus[A,A]={A}.

A

B

Figure 2.3. A segment[A,B].

Segments have the following properties:-

(i)If A=B,then[A,B]⊂AB.

(ii) A,B∈[A,B]for all A,B∈Π.

(iii) [A,B]=[B,A]for all A,B∈Π.

(iv) If C,D∈[A,B]then[C,D]⊂[A,B].

(v)If A,B,C are distinct points on a line l,then precisely one of

A∈[B,C],B∈[C,A],C∈[A,B],

holds.

Proof. In each case we suppose thatA≤lBso that we have (2.1.1) above; other-
wise replace≤lby≥lthroughout to cover (2.1.2).
(i) By A 1 ,l=ABso[A,B]is a set of points onAB.
(ii) By A 2 (i)A≤lA≤lBandA≤lB≤lB.
(iii) AsA≤lB,thenB≥lAso[B,A]={P∈l:B≥lP≥lA}.NowifP∈[A,B],
thenA≤lPandP≤lB. It follows thatB≥lPandP≥lA. ThusP∈[B,A]and so
[A,B]⊂[B,A]. By a similar argument[B,A]⊂[A,B]and so[A,B]=[B,A].