214 Vector and complex-number methods Ch. 11
Assumingq 1 =0 we have the pair of equations
1
√
p^21 +q^21
r−
1
√
( 1 −p 1 )^2 +q^21
s= 0 ,
⎛
⎝ 1 +√ p^1
p^21 +q^21
⎞
⎠r+
⎛
⎝ 1 +√^1 −p^1
( 1 −p 1 )^2 +q^21
⎞
⎠s= 2 |Z 2 ,Z 3 |. (11.6.8)
We denote byD 1 the value of the determinant of coefficients on the left-hand side
∣∣
∣∣
∣
∣
√^1
p^21 +q^21 −
√^1
( 1 −p 1 )^2 +q^21
1 +√pp 21
1 +q^21
1 +√( 11 −−pp^1
1 )^2 +q^21
∣∣
∣∣
∣
∣
and have
D 1 =
√
( 1 −p 1 )^2 +q^21 +
√
p^21 +q^21 + 1
√
p^21 +q^21
√
( 1 −p 1 )^2 +q^21
.
We denote byD 2 the value of the determinant
∣∣
∣∣
∣∣
0 −√( 1 −p^1
1 )^2 +q^21
2 |Z 2 ,Z 3 | 1 +√( 11 −−pp^1
1 )^2 +q^21
∣∣
∣∣
∣∣
which has the value
D 2 =
√^2 |Z^2 ,Z^3 |
( 1 −p 1 )^2 +q^21
.
Then we have the solution
r=
D 2
D 1
=
2 |Z 2 ,Z 3 |
√
p^21 +q^21
√
( 1 −p 1 )^2 +q^21
√
p^21 +q^21 + 1
.
Thus the point of intersection from this bisector is
z 2 +
√
p^21 +q^21
√
( 1 −p 1 )^2 +q^21 +
√
p^21 +q^21 + 1
⎡
⎣ 1 +√p^1 +ıq^1
p^21 +q^21
⎤
⎦(z 3 −z 2 ), (11.6.9)
which is thus the incentreZ 15.