226 Vector and complex-number methods Ch. 11
11.7 Somewell-knowntheorems......................
NOTE. The advantage of mobile coordinates is that they located a point with respect
to a triangle using just two instead of three numbers and they also behave like re-
scaled Cartesian coordinates. None the less they can lead to unwieldy expressions as
in this section and it is a good idea when possible to check the algebraic manipula-
tions using a computer software programme.
11.7.1 Feuerbach’s theorem, 1822 ......................
We recall that the nine-point circle has radius-length equal to the distance fromZ′ 16
toZ 4. From the formula
z′ 16 −z 4 =^12 (p 1 +q 1 ı)[
1 −
ı
2 q 1(p 1 − 1 −q 1 ı)]
(z 3 −z 2 )=
1
4 q 1(p 1 +q 1 ı)[q 1 +( 1 −p 1 )ı](z 3 −z 2 )we note that
|z′ 16 −z 4 |=a
4 |q 1 ||p 1 +q 1 ı||q 1 +( 1 −p 1 )ı|=a
4 |q 1 |√
p^21 +q^21√
(p 1 − 1 )^2 +q^21 ,and sothis is the radius-length of the nine-point circle.We denote this radius length
byr 1.
Our formula in 11.6.5 for the incentre of a triangle is very awkward to apply
because of the complicated term in the denominator. However by eliminating the
surds in the denominator in two steps, by multiplying above and below by a conjugate
surd of the denominator, we obtain the more convenient formulation that
p 1 +√
p^21 +q^211 +√
p^21 +q^21 +√
(p 1 − 1 )^2 +q^21=
1
2
+
1
2
√
p^21 +q^21 −1
2
√
(p 1 − 1 )^2 +q^21 .(11.7.1)In fact once we know the form of this we can establish it more directly and easily by
noting that
1
2[
1 +
√
p^21 +q^21 +√
(p 1 − 1 )^2 +q^21][
1 +
√
p^21 +q^21 −√
(p 1 − 1 )^2 +q^21]
=^12
[(
1 +
√
p^21 +q^21) 2
−
(
(p 1 − 1 )^2 +q^21)
]
=p 1 +√
p^21 +q^21.We note that the right-hand side in (11.7.1) must be positive.