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 ].