Sec. 3.3 A ratio result 39
(ii)For all A,B∈Π,mp(A,B)∈[A,B].
(iii)In all cases mp(A,A)=A, and mp(A,B)=A, mp(A,B)=BwhenA=B.
(iv)Given any points P and Q inΠ, there is a unique point R∈Πsuch that Q=
mp(P,R).
Proof.
(i) WhenA=Bthis follows from the definition and A 4 (ii); whenA=Bit is
immediate.
(ii) WhenA=Bthis follows from the preparatory result. WhenA=Bit amounts
toA∈{A}.
(iii) This follows from the definition and preparatory result.
(iv)Existence.IfQ=Pwe takeR=Pandthenmp(P,R)=mp(P,P)=P=Q.
Suppose then thatP=Q,letl=PQand let≤lbe the natural order onlunder which
P≤lQ.TakeRonlso thatP≤lRand|P,R|= 2 |P,Q|.ThenPprecedes bothQandR
onl, while|P,Q|=^12 |P,R|. By our initial specification ofXin the preparatory result
we see thatQ=mp(P,R).
Uniqueness. Suppose that alsoQ=mp(P,S). We again first takeQ=P.Nowin
the preparatory result we hadX=A, so that cannot be the situation here asQ=P;
thus we must haveS=Pand soS=R. Next suppose thatQ=P; then we cannot
haveS=P,aswehadX=A.ThenQ∈PS,sobyA 1 S∈PQ. In factQ∈[P,S]so
asP≤lQwe must haveP≤lS; moreover|P,R|=|P,S|as each is twice the distance
|P,Q|. By the uniqueness in A 4 (iv) we must then haveR=S.
3.3 Aratioresult..............................
3.3.1
Let A,B,C be distinct collinear points, and write
|A,C|
|A,B|
=r,
|A,C|
|C,B|
=s.
Then if C∈[A,B]we have
s=
r
1 −r
,r=
s
1 +s
.
Proof.Let|A,C|=x,|C,B|=y.AsC∈[A,B]we have|A,B|=x+y.Then
x
x+y
=r so that
x+y
x
=
1
r
.
Hence
y
x
=^1
r
−1andsox
y
=|A,C|
|C,B|
= r
1 −r