Sec. 7.6 Sensed distances 117
7.6.2 Sensed products and a circle......................
The conclusion of 7.4.2 can be strengthened to replace|P,R||P,S|byPRPS. In fact
the initial analytic proof gives this but it also easily follows from the stated result as
PRPS=−|P,R||P,S|whenPis interior to the circle whilePRPS=|P,R||P,S|when
Pis exterior to the circle. We now look to a converse type of result.
Suppose that Z 1 ,Z 2 and Z 3 are fixed non-collinear points. For a variable point W
let Z 1 W meet Z 2 Z 3 at W′and
W′WW′Z 1 =W′Z 2 W′Z 3.Then W lies on the circle which passes through Z 1 ,Z 2 and Z 3.
Proof. Without loss of generality
we may take our frame of refer-
ence so thatZ 1 ≡( 0 ,y 1 ), Z 2 ≡
(x 2 , 0 ), Z 3 ≡(x 3 , 0 ),andwe
takeW≡(u,v),W′≡(u′, 0 ).
Z 1
Z 2 Z 3
W′
W
Figure 7.9.
Then it is easily found thatu′=y 1 u/(y 1 −v), and so, first of all,W′Z 2 W′Z 3 =(
x 2 −y 1 u
y 1 −v)(
x 3 −y 1 u
y 1 −v)
.
The lineW′Whas parametric equationsx=u′+s(u−u′),y= 0 +s(v− 0 ), withs= 0
givingW′ands=1givingW. ThusW′W=|W′,W|. The pointZ 1 has parameter
given byy 1 =svand sos=y 1 /v;thenW′Z 1 =yv^1 |W′,W|. It follows that
W′WW′Z 1 =
y 1
v|W′,W|^2 =
y 1
v[(
u−y 1 u
y 1 −v) 2
+v^2]
=y 1 v[(
u
y 1 −v) 2
+ 1
]
.
On equating the two expressions we have
(
x 2 −
y 1 u
y 1 −v)(
x 3 −y 1 u
y 1 −v)
=y 1 v[(
u
y 1 −v) 2
+ 1
]
,
which we re-write as
y 1 u^2 (y 1 −v)
(y 1 −v)^2=y 1 v−x 2 x 3 +y 1 (x 2 +x 3 )u
y 1 −v.
On multiplying across byy 1 −vwe obtain
y 1 (u^2 +v^2 )−y 1 (x 2 +x 3 )u−(y^21 +x 2 x 3 )v+y 1 x 2 x 3 = 0 ,and this is the equation of a circle.