66 The parallel axiom; Euclidean geometry Ch. 5
Let A,B,C be non-collinear points and let P∈[A,B and Q∈[A,C be such that
PQ‖BC. Then
|A,P|
|A,B|
=
|A,Q|
|A,C|
.
A
B C
P Q
B 1
B 2
B 4
B 5
B 6
B 7
B 8
B 9
C 1
C 2
C 4
C 5
C 6
C 7
C 8
C 9
Figure 5.6.
A
B C
P Q
Pu
Pv
Qu
Qv
Proof. We assume first thatP∈[A,B]. Within this first case, we suppose initially
that
|A,P|
|A,B|
=
s
t
,
wheresandtare positive whole numbers withs<t,sothats/tis an arbitrary rational
number between 0 and 1. For 0≤j≤tletBjbe the point on[A,B such that
|A,Bj|
|A,B|
=
j
t
,
so thatB 0 =A,Bt=BandBs=P.IfAB=land≤lis the natural order for which
A≤lB,thenA≤lBj− 1 ≤lBj≤lBj+ 1 ≤lBand|Bj− 1 ,Bj|=|Bj,Bj+ 1 |.IfAC=m
and≤mis the natural order for whichA≤mC, then by the last result applied with
(D 1 ,D 2 ,D 3 )=(Bj− 1 ,Bj,Bj+ 1 ),for1≤j≤t−1 the line throughBjwhich is parallel
toBCwill meetACin a pointCjsuch thatA≤mCj− 1 ≤mCj≤mCj+ 1 ≤mCand
|Cj− 1 ,Cj|=|Cj,Cj+ 1 |.
It follows that, for 0≤j≤t,|A,Cj|=j|A,C 1 |andsoasCt=C,
|A,Cj|
|A,C|
=
j|A,C 1 |
t|A,C 1 |
=
j
t
.
In particular, asCs=Q, it follows that
|A,Q|
|A,C|
=
s
t