38 Distance; degree-measure of an angle Ch. 3
A=C
D
B
Figure 3.3.A
C
B
DX
3.2 Mid-points...............................
3.2.1
If A=B there is a unique point X on l=AB such that|A,X|=|X,B|.Inthisinfact
X∈[A,B]and X=A,X=B.
Proof.
Existence.Let≤lbe the natural order onlfor whichA≤lB. Withk=^12 |A,B|,by
A 4 (iv) there is a pointXonlsuch thatA≤lXand|A,X|=^12 |A,B|. ClearlyX∈[A,B.
As|A,B|>0wehave|A,X|<|A,B|; by 3.1.2 this implies thatX∈[A,B],X=B.By
A 4 (iii)|A,X|+|X,B|=|A,B|and so|X,B|=|A,B|−^12 |A,B|=^12 |A,B|. Thus|A,X|=
|X,B|as required. We have already seen thatX∈[A,B]andX=B;as|A,X|>0we
also haveX=A.
Uniqueness. Suppose now thatY∈land|A,Y|=|Y,B|.ThenYcannot beAor
B,ase.g.Y=Aimplies that|A,A|=|A,B|,i.e.0=|A,B|. Thus by 2.1.4 one of
Y∈[A,B],A∈[Y,B],B∈[A,Y],holds. The second of these would imply|Y,A|+|A,B|=|Y,B|and so|Y,A|<|Y,B|
as|A,B|>0. The third of these would imply|A,B|+|B,Y|=|A,Y|and so|B,Y|<
|A,Y|. As these contradict our assumptions, we must haveY∈[A,B].Then|A,Y|+
|Y,B|=|A,B|and as|A,Y|=|Y,B|this implies that|A,Y|=^12 |A,B|.ThenA≤l
X,A≤lYand|A,X|=|A,Y|so by the uniqueness in A 4 (iv) we must haveX=Y.
Definition. Given any points
A,B∈Π,wedefinethemid-
pointofAandBas follows: if
A=Bthen the mid-point isA;
whenA=Bthe mid-point is the
unique pointX on the lineAB
such that|A,X|=|X,B|,which
has just been guaranteed. We de-
note the mid-point ofAandBby
mp(A,B).
A
X
B
++Figure 3.4. Mid-point ofAandB.Mid-points have the following properties:-(i)For all A,B∈Π,mp(A,B)=mp(B,A).