Sec. 6.7 Coordinate treatment of harmonic ranges 99x 4 =x 1 +λ
λ− 1(x 2 −x 1 ),y 4 =y 1 +λ
λ− 1(y 2 −y 1 ),Nowλ/(λ− 1 )>1andsoλ>1. Hence^12 <λ/( 1 +λ)<1, and soZ 3 ∈[Z 1 ,Z 2 ].
ThusZ 2 ,Z 3 andZ 4 are on the one side ofZ 1 on the lineZ 1 Z 2 .Then|Z 1 ,Z 3 |
|Z 1 ,Z 2 |=
λ
λ+ 1,
|Z 1 ,Z 4 |
|Z 1 ,Z 2 |
=
λ
λ− 1,
so that
|Z 1 ,Z 2 |
|Z 1 ,Z 3 |=
λ+ 1
λ,
|Z 1 ,Z 2 |
|Z 1 ,Z 4 |
=
λ− 1
λ,
and so
|Z 1 ,Z 2 |
|Z 1 ,Z 3 |+
|Z 1 ,Z 2 |
|Z 1 ,Z 4 |
=
λ+ 1
λ+
λ− 1
λ= 2.
Hence
1
2(
1
|Z 1 ,Z 3 |
+
1
|Z 1 ,,Z 4 |
)
=
1
|Z 1 ,Z 2 |
This is expressed by saying that|Z 1 ,Z 2 |is theharmonic meanof|Z 1 ,Z 3 |and
|Z 1 ,Z 4 |.6.7.5 Constructionforaharmonicrange
Z 1
W 3
Z 3 Z 2
Z 4
W 1
l W 2Figure 6.7.Let Z 1 ,Z 2 ,Z 3 be distinct collinear points with Z 3 not the mid-point of Z 1 and Z 2 .Take
any points W 1 and W 2 , not on Z 1 Z 2 , so that Z 2 is the mid-point of W 1 and W 2 .Letl
be the line through Z 1 which is parallel to W 1 W 2 and let W 3 be the point in which
W 1 Z 3 meets l, with Z 4 the point in which W 2 W 3 meets Z 1 Z 2 .Then(Z 1 ,Z 2 ,Z 3 ,Z 4 )is a
harmonic range.
Proof. Without loss of generality we may take thex-axis to be the lineZ 1 Z 2 and
so take coordinates
Z 1 ≡(x 1 , 0 ),Z 2 ≡(x 2 , 0 ),Z 3 ≡(x 3 , 0 ),Z 4 ≡(x 4 , 0 ),andW 1 ≡(u 1 ,v 1 ),W 2 ≡( 2 x 2 −u 1 ,−v 1 ). The lineslandW 1 Z 3 have equations(u 1 −x 2 )y=v 1 (x−x 1 ), (u 1 −x 3 )y=v 1 (x−x 3 ),