Sec. 11.7 Some well-known theorems 227
Recalling from 11.6.5 that the incentreZ 15 has complex coordinatez 15 =z 2 +p 1 +√
p^21 +q^21 +q 1 ı1 +√
p^21 +q^21 +√
(p 1 − 1 )^2 +q^21(z 3 −z 2 ),we re-write this as
z 15 −z 2=
1
2
[
1 +
√
p^21 +q^21 −√
(p 1 − 1 )^2 +q^21]⎡
⎣ 1 + q^1 ı
p 1 +√
p^21 +q^21⎤
⎦(z 3 −z 2 )=
1
2 q 1[
1 +
√
p^21 +q^21 −√
(p 1 − 1 )^2 +q^21][
q 1 +(√
p^21 +q^21 −p 1)
ı]
(z 3 −z 2 ).From this, the footZ 20 of the perpendicular from the incentreZ 15 to the lineZ 2 Z 3 has
complex coordinate
z 20 =z 2 +1
2 q 1[
1 +
√
p^21 +q^21 −√
(p 1 − 1 )^2 +q^21]
q 1 (z 3 −z 2 ) (11.7.2)and sothe length of radius of the incircle is equal to
a
2[
1 +
√
p^21 +q^21 −√
(p 1 − 1 )^2 +q^21][√
p^21 +q^21 −p 1]
. (11.7.3)
We denote this radius length byr 2.
With these preparatory re-
sults, we can now show
thatthe nine-point circle
and the incircle meet at
just one point and they
have a common tangent
there.
Z^15Z 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 11.17. Feuerbach’s theorem.
It is clear from diagrams thatr 1 >r 2 and known from earlier proofs. We will first
give the proof in this case and then a proof thatr 4 ≥r 1 cannot occur.
The half-line[Z 16 ′,Z 15 has pointsZ=Z′ 16 +s(Z 15 −Z 16 ′)wheres≥0 and will
meet the nine-point circle at a point
z 21 =z′ 16 +r 1
|z 15 −z′ 16 |(z 15 −z′ 16 ).