100 Cartesian coordinates; applications Ch. 6
respectively, and soW 3 has coordinates
u 3 =
x 3 (u 1 −x 2 )−x 1 (u 1 −x 3 )
x 3 −x 2
, v 3 =v 1
x 3 −x 1
x 3 −x 2
.
On forming the equation ofW 2 W 3 and finding where it meetsZ 1 Z 2 we obtain
x 4 =
−x 3 (x 1 +x 2 )+ 2 x 1 x 2
x 1 +x 2 − 2 x 3
,
from which it follows that
x 4 −x 1 =
(x 1 −x 2 )(x 3 −x 1 )
x 1 +x 2 − 2 x 3
, x 2 −x 4 =
(x 1 −x 2 )(x 3 −x 2 )
x 1 +x 2 − 2 x 3
.
From these we see that
x 4 −x 1
x 2 −x 4
=−
x 3 −x 1
x 2 −x 3
.
Exercises
6.1 Suppose thatZ 1 ,Z 2 ,Z 3 are non-collinear points andZ 5 =mp{Z 3 ,Z 1 },
Z 6 =mp{Z 1 ,Z 2 }. Show that if|Z 2 ,Z 5 |=|Z 3 ,Z 6 |,then|Z 3 ,Z 1 |=|Z 1 ,Z 2 |.
(Hint. Select a frame of reference to simplify the calculations).
6.2 Letl 1 ,l 2 be distinct intersecting lines andZ 0 a point not on either of them.
Show that there are unique pointsZ 1 ∈l 1 ,Z 2 ∈l 2 such thatZ 0 is the mid-point
ofZ 1 andZ 2.
6.3 Suppose thatZ 1 ,Z 2 ,Z 3 are non-collinear points. Show that the pointsZ≡(x,y),
the perpendicular distances from which to the linesZ 1 Z 2 ,Z 1 Z 3 are equal, are
those the coordinates of which satisfy
−(y 2 −y 1 )(x−x 1 )+(x 2 −x 1 )(y−y 1 )
√
(x 2 −x 1 )^2 +(y 2 −y 1 )^2
±
−(y 3 −y 1 )(x−x 1 )+(x 3 −x 1 )(y−y 1 )
√
(x 3 −x 1 )^2 +(y 3 −y 1 )^2
= 0.
Show that ifx=( 1 −t)x 2 +tx 3 ,y=( 1 −t)y 2 +ty 3 thenZlies on the line with
equation
−(y 2 −y 1 )(x−x 1 )+(x 2 −x 1 )(y−y 1 )
√
(x 2 −x 1 )^2 +(y 2 −y 1 )^2
+
−(y 3 −y 1 )(x−x 1 )+(x 3 −x 1 )(y−y 1 )
√
(x 3 −x 1 )^2 +(y 3 −y 1 )^2
= 0 ,
if and only
t=
|Z 1 ,Z 2 |^2
|Z 1 ,Z 2 |^2 +|Z 2 ,Z 3 |^2
.
Deduce that this latter line is the mid-line of|Z 2 Z 1 Z 3.