Sec. 11.7 Some well-known theorems 233
we havew 3 =z 2 +(p+^12 +ıq′)(z 3 −z 2 ), for some real numberq′.ButZ 4 ,W 2 ,W 3 are
collinear, so that ∣
∣∣
∣∣
∣
2 p 01
pq 1
p+^12 q′ 1∣∣
∣∣
∣
∣
= 0 ,
and from thisq′=p−
(^12)
p qı(z^3 −z^2 ). Hencew^3 =z^2 +
(
p+^12 +p−(^12)
p qı
)
(z 3 −z 2 ).Fromthisw 3 −z 3 =p−
(^12)
p (p+qı)(z^3 −z^2 ), and sincez^3 −z^2 =
1
p 1 − 1 +ıq 1 (z^1 −z^3 ),wehave
w 3 =z 3 +p−
(^12)
p
(p+ıq)(p 1 − 1 −ıq 1 )
(p 1 − 1 )^2 +q^21 (z^1 −z^3 ). Thus ifZ^9 is the foot of the perpendicular
fromW 3 toZ 3 Z 1 ,wehavez 9 =z 3 +p−
(^12)
p
p(p 1 − 1 )+qq 1
(p 1 − 1 )^2 +q^21 (z^1 −z^3 ),andso
z 5 =z 3 +
2 p− 1
p
p(p 1 − 1 )+qq 1
(p 1 − 1 )^2 +q^21
(z 1 −z 3 ).
Similarlyw 2 =z 2 +(p+ıq)(z 3 −z 2 ),z 3 −z 2 =p 1 +^1 ıq 1 (z 1 −z 2 ),andsow 2 =z 2 +
(p+ıq)(p 1 −ıq 1 )
p^21 +q^21 (z^1 −z^2 ). It follows that for the footZ^10 of the perpendicular fromW^2 to
Z 1 Z 2 we havez 10 =z 2 +ppp^12 +qq^1
1 +q^21
(z 1 −z 2 ),andsoz 6 =z 2 + 2 ppp^12 +qq^1
1 +q^21
(z 1 −z 2 ).
We have
z 1 −z 5 =
[
1 −
2 p− 1
pp(p 1 − 1 )+qq 1
(p 1 − 1 )^2 +q^21]
(z 1 −z 3 ),z 1 −z 6 =[
1 − 2
pp 1 +qq 1
p^21 +q^21]
(z 1 −z 2 ),z 4 −z 6 = 2 p(z 3 −z 2 )− 2pp 1 +qq 1
p^21 +q^21(z 1 −z 2 )= 2
[
p−pp 1 +qq 1
p^21 +q^21(p 1 +ıq 1 )]
(z 3 −z 2 ),z 4 −z 5 =[
( 2 p− 1 )(z 3 −z 2 )−2 p− 1
pp(p 1 − 1 )+qq 1
(p 1 − 1 )^2 +q^21(z 1 −z 3 )]
=
2 p− 1
p[
p−p(p 1 − 1 )+qq 1
(p 1 − 1 )^2 +q^21(p 1 − 1 +q 1 )]
(z 3 −z 2 ).Asz 1 −z 2 =(p 1 +ıq 1 )(z 3 −z 2 ), from these combined we have that((zz^11 −−zz^56 )()(zz 44 −−zz^65 ))is a
real multiple of
[
p−ppp 21 +qq^1
1 +q^21
(p 1 +ıq 1 )]
(p 1 − 1 +ıq 1 )
[p−p((pp 11 −− 11 ))+ (^2) +qqq 21
1
(p 1 − 1 +ıq 1 )
]
(p 1 +ıq 1 )=
[
p−ppp 11 −+ıqqq 11]
(p 1 − 1 +ıq 1 )
[
p−p(pp 11 −− 11 −)+ıqqq 11]
(p 1 +ıq 1 )=−q 1 (ıp+q)[(p 1 − 1 )^2 +q^21 ]
−q 1 (ıp+q)[p^21 +q^21 ]=
(p 1 − 1 )^2 +q^21
p^21 +q^21,
and this is real. It follows thatZ 1 ,Z 4 ,Z 5 ,Z 6 are concyclic. This is known as
Miquel’s theorem.