Geometry with Trigonometry

Sec. 11.6 Mobile coordinates 223

Nowz 4 =z 2 +^12 (z 3 −z 2 ),z 8 =z 2 +p 1 (z 3 −z 2 ), and by the formula in 11.6.5 for

the orthocentre,z 11 =z 2 +p 21

(

1 −p^1 q− 11 ı

)

(z 3 −z 2 ). From these we have that

z 18 =^12 [z 2 +z 2 +p 1

(

1 −

p 1 − 1

q 1

ı

)

(z 3 −z 2 )=z 2 ++

1

2

p 1 ( 1 −

p 1 − 1

q 1

ı)(z 3 −z 2 ).

We letZ 16 (iii)be the circumcentre of the triangle[Z 4 ,Z 8 ,Z 18 ]and for the present write

z 4 =z 8 +(p 4 +ıq 4 )(z 18 −z 8 )=( 1 −p 4 −ıq 4 )z 8 +(p 4 +ıq 4 )z 18

=( 1 −p 4 −ıq 4 )[z 2 +p 1 (z 3 −z 2 )]+ (p 4 +ıq 4 )[z 2 +

1

2

p 1 ( 1 −

p 1 − 1

q 1

ı)](z 3 −z 2 )

=( 1 −ıq 4 +ıq 4 )z 2 +

[

( 1 −p 4 −ıq 4 )p 1 +(p 4 +ıq 4 )

1

2

p 1 ( 1 −

p 1 − 1

q 1

)ı

]

(z 3 −z 2 )

=z 2 +p 1

[

1 −p 4 −ıq 4 +^12 (p 4 +ıq 4 )[ 1 −

p 1 − 1

q 1

ı]

]

(z 3 −z 2 )

=z 2 +p 1

[

1 −p 4 −q 4 ı+^12 (p 4 +q 4 ı)−^12 (p 4 +q 4 ı)

p 1 − 1

q 1

ı

]

(z 3 −z 2 )

=z 2 +p 1

[

1 −^12 p 4 −^12 q 4 ı−^12 p 4

p 1 − 1

q 1

ı+^12 q 4

p 1 − 1

q 1

]

(z 3 −z 2 )

=z 2 +p 1

[

1 −^12 p 4 +^12 q 4

p 1 − 1

q 1

−^12 (q 4 +p 4

p 1 − 1

q 1

)ı

]

(z 3 −z 2 )

=z 2 +^12 (z 3 −z 2 ).

Hence

p 1

[

1 −^12 p 4 +^12 q 4

p 1 − 1

q 1

−^12 (q 4 +p 4

p 1 − 1

q 1

)ı

]

=^12

so that

q 4 +p 4

p 1 − 1

q 1

= 0 ,p 1

[

1 −^12 p 4 +^12 q 4

p 1 − 1

q 1

]

=

1

2

.

Thenq 4 =−p^1 q− 11 p 4 and

−^12 p 1 p 4 +^12

p 1 − 1

q 1

q 4 =

1

2

−p 1 ,−p 4 +

p 1 − 1

q 1

q 4 =(^12 −p 1 )

2

p 1

,

−p 4 −

p 1 − 1

q 1

p 1 − 1

q 1

p 4 =

1

p 1

− 2 ,−p 4

[

1 +

(p 1 − 1 )^2

q^21

]

=

1 − 2 p 1

p 1

,

p 4 =

( 2 p 1 − 1 )q^21

p 1 (q^21 +(p 1 − 1 )^2 )

,

q 4 =−

p 1 − 1

q 1

2 p 1 − 1

p 1

q^21

[q^21 +(p 1 − 1 )^2 ]

p^24 +q^24 =

[

1 +(

p 1 − 1

p 1

)^2

]

( 2 p 1 − 1 )^2

p^21

q^41

[q^21 +(p 1 − 1 )^2 ]

.