98 Cartesian coordinates; applications Ch. 6
then
x 1 =
1
1 +μ
x 3 +
μ
1 +μ
x 4 ,y 1 =
1
1 +μ
y 3 +
μ
1 +μ
y 4.
If we defineμ′by
1
1 +μ′
=
1 +λ
2 λ
,
so that
μ′=
λ− 1
1 +λ
,
μ′
1 +μ′
=
λ− 1
2 λ
,
then
x 2 =
1
1 +μ′
x 3 +
μ′
1 +μ′
x 4 ,y 2 =
1
1 +μ′
y 3 +
μ′
1 +μ′
y 4.
Asμ′=−μ, this shows thatZ 1 andZ 2 divide{Z 3 ,Z 4 }internally and externally in
the same ratio.
6.7.3 Distancesfrommid-point
Let Z 0 be the mid-point of distinct points Z 1 and Z 2. Then points Z 3 ,Z 4 ∈Z 1 Z 2 divide
{Z 1 ,Z 2 }internally and externally in the same ratio if and only if Z 3 and Z 4 are on the
one side of Z 0 on the line Z 1 Z 2 and
|Z 0 ,Z 3 ||Z 0 ,Z 4 |=^14 |Z 1 ,Z 2 |^2.
Proof.WehaveZ 0 ≡(x 0 ,y 0 )wherex 0 =^12 (x 1 +x 2 ),y 0 =^12 (y 1 +y 2 ).Then
x 3 −x 0 =(t−^12 )(x 2 −x 1 ), y 3 −y 0 =(t−^12 )(y 2 −y 1 ),
x 4 −x 0 =(s−^12 )(x 2 −x 1 ), y 4 −y 0 =(s−^12 )(y 2 −y 1 ),
and so
|Z 0 ,Z 3 ||Z 0 ,Z 4 |=|
(
t−^12
)(
s−^12
)
||Z 1 ,Z 2 |^2.
By (6.7.1)Z 3 ,Z 4 divide{Z 1 ,Z 2 }internally and externally in the same ratio if and
only if
(
s−^12
)(
t−^12
)
=^14. This is equivalent to having|
(
s−^12
)(
t−^12
)
( |=^14 and
s−^12
)(
t−^12
)
- The latter is equivalent to having eithers−^12 >0andt−^12 >0,
ors−^12 <0andt−^12 <0, so thatZ 3 andZ 4 are on the one side ofZ 0 on the line
Z 1 Z 2.
6.7.4 Distancesfromend-point
Let{Z 3 ,Z 4 }divide{Z 1 ,Z 2 }internally and externally in the same ratio with Z 2 ∈
[Z 1 ,Z 4 ].Then
1
2
(
1
|Z 1 ,Z 3 |
+
1
|Z 1 ,Z 4 |
)
=
1
|Z 1 ,Z 2 |
.
Proof.Wehaveasbefore
x 3 =x 1 +
λ
1 +λ
(x 2 −x 1 ),y 3 =y 1 +
λ
1 +λ
(y 2 −y 1 ),