Geometry with Trigonometry

Sec. 11.6 Mobile coordinates 221

Then

p^22 +q^22 =

( 2 p 1 − 1 )^2 (p^21 +q^21 )

(p^21 +q^21 )^2

=

( 2 p 1 − 1 )^2

p^21 +q^21

.

By (11.6.4) the circumcentreZ 16 of the triangle[Z 1 ,Z 2 ,Z 3 ]has complex coordi-
nate

z 16 =z 2 +^12

[

1 −

p 1

q 1

ı+

p^21 +q^21

q 1

ı

]

(z 3 −z 2 ),

and so the circumcentreZ( 16 ii)of of the triangle[Z 4 ,Z 8 ,Z 6 ]has complex coordinate

z( 16 ii)=z 8 +^12

[

1 −

p 2

q 2

ı+

p^22 +q^22

q 2

ı

]

(z 6 −z 8 )

=z 2 +p 1 (z 3 −z 2 )+^12

[

1 −

p 2

q 2

ı+

p^22 +q^22

q 2

ı

](

1

2 (p^1 +q^1 ı)(z^3 −z^2 )−p^1 (z^3 −z^2 )

)

=z 2 +

[

p 1 +^12 ( 1 −

p 2

q 2

ı+

p^22 +q^22

q 2

ı)^12 (−p 1 +q 1 ı)

]

(z 3 −z 2 )

=z 2 +p 1 (z 3 −z 2 )+

+^14

[

−p 1 +

p 2

q 2

q 1 −q 1

p^22 +q^22

q 2

+[p 1

p 2

q 2

−p 1

p^22 +q^22

q 2

+q 1 ]ı

]

(z 3 −z 2 ). (11.6.18)

From the values ofp^22 +q^22 andq 2 , and also ofp 2 /q 2 we have−q 1 p

(^22) +q (^22)
q 2 +q^1
p 2
q 2 =
−q 1 (^2 p^1 −^1 )
2
p^21 +q^21
p^21 +q^21
q 1 ( 2 p 1 − 1 )+q^1
p 1
q 1 =−(^2 p^1 −^1 )+p^1 =^1 −p^1.
Similarly
p 1
p 2
q 2
−p 1
p^22 +q^22
q 2
=−p 1
( 2 p 1 − 1 )^2
p^21 +q^21
p^21 +q^21
( 2 p 1 − 1 )q 1
+p 1
p 1
q 1
=−
p 1
q 1
( 2 p 1 − 1 )+p 1
p 1
q 1
=− 2 p 1
p 1
q 1

+

p 1

q 1

+p 1

p 1

q 1

=( 1 −p 1 )

p 1

q 1

.

Hence

z( 16 ii)=z 2 +p 1 (z 3 −z 2 )+^14

[

−p 1 + 1 −p 1 +[( 1 −p 1 )

p 1

q 1

+q 1 ]ı

]

(z 3 −z 2 )

=z 2 +^14

[

4 p 1 + 1 − 2 p 1 +[

p 1

q 1

−

p^21 −q^21

q 1

]ı

]

(z 3 −z 2 )

=z 2 +^12

[

1

2 +p^1 +

1

2 [

p 1

q 1

−

p^21 +q^21

q 1

+ 2 q 1 ]ı

]

(z 3 −z 2 ). (11.6.19)

This is the same as (11.6.16) and soz( 16 ii)=z′ 16.
The reason we took the particular foot of a perpendicularZ 8 is it simplify the
calculations a little, but the result holds just as well for the feet of the perpendicu-
larsZ 9 andZ 10 as well. For we can give the simplifying to both of these as well as