Geometry with Trigonometry

222 Vector and complex-number methods Ch. 11

follows. Starting with our usual notationz 1 =z 2 +(p 1 +ıq 1 )(z 3 −z 2 )let us seek a
corresponding relationshipz 3 =z 1 +(p 3 +ıq 3 )(z 2 −z 1 )based on the same triangle.
Then we have

(p 3 +q 3 ı− 1 )z 1 =−z 3 +(p 3 +ıq 3 )z 2 =(p 3 +ıq 3 − 1 )z 2 −(z 3 −z 2 ),

z 1 =z 2 +

− 1

p 3 +q 3 ı− 1

(z 3 −z 2 )=z 2 +(p 1 +q 1 ı)(z 3 −z 2 ),

p 1 +q 1 ı= −^1

p 3 +q 3 ı− 1

,

p 3 +q 3 ı− 1 =

− 1

p 1 +q 1 ı

p 1 −q 1 ı

p 1 −q 1 ı

=

−p 1 +q 1 ı

p^21 +q^21

p 3 − 1 =

−p 1

p^21 +q^21

,

q 3 =

p 1

p^21 +q^21

,

p 3 = 1 −

p 1

p^21 +q^21

=

p^21 +q^21 −p 1

p^21 +q^21

. (11.6.20)

11.6.11Thenine-pointcircle.........................

Z 11

Z 1

Z 2

Z 3

Z 16 ′

Z 4

Z 5

Z 6

Z 8

Z 9

Z 10 =Z 18

Z 17

Z 19

Figure 11.16. Nine-point circle.

We continue with the situation in 11.6.10 where we identified three further points
which lie on the circumcircle of the triangle[Z 4 ,Z 5 ,Z 6 ]with vertices the midpoints
Z 4 ,Z 5 andZ 6 of the sides[Z 2 ,Z 3 ],[Z 3 ,Z 1 ],[Z 1 ,Z 2 ], respectively, of the original tri-
angleZ 1 ,Z 2 ,Z 3. In this subsection we identify three other points which lie on this
circumcircle. LetZ 17 be the mid-point of the orthocentre and the vertexZ 1 in the
original triangle,Z 18 be the mid-point of the orthocentre and the vertexZ 2 ,andZ 19
be the mid-point of the orthocentre and the vertexZ 3. We seek the circumcentre of
the triangle[Z 4 ,Z 8 ,Z 18 ].