Sec. 11.7 Some well-known theorems 231
11.7.3 The incentre on the Euler line of a triangle ...........
We suppose that we have the mobile coordinatesz 1 −z 2 =(p 1 +q 1 ı)(z 3 −z 2 ),where
p 1 andq 1 are real numbers andq 1 =0. Thenz 1 −z 3 =(p 1 − 1 +q 1 ı)(z 3 −z 2 ),andas
in 11.6.3 we have
z 1 −z 2
z 3 −z 2
=p 1 +q 1 ı=c
a
cisβ, z^2 −z^3
z 1 −z 3
=^1 −p^1 +q^1 ı
( 1 −p 1 )^2 +q^21
=a
b
cisγ,
z 3 −z 1
z 2 −z 1
=
p 1 − 1 +q 1 ı
p 1 +q 1 ı
=
b
c
cisα,
where we are using our standard notation. Thenp^21 +q^21 =c
2
a^2 ,(^1 −p^1 )
(^2) +q 2
1 =
b^2
a^2.
We recall that the orthocentre, centroid and incentre have mobile coordinatesp 1 +
p 1 ( 1 −p 1 )
q 1 ı,
p 1 + 1
3 +
q 1
3 ı,
p 1 +
√
p^21 +q^21 +q 1 ı
1 +
√
p^21 +q^21
√
( 1 −p 1 )^2 +q^21 respectively.
Now 1− 2 p 1 =b
(^2) −c 2
a^2 ,p^1 =
c^2 +a^2 −b^2
2 a^2 ,^1 −p^1 =
a^2 +b^2 −c^2
2 a^2 ,p^1 +^1 =
c^2 +a^2 −b^2 + 2 a^2
2 a^2.
Moreover
q^21 =
c^2
a^2
−p^21 =
c^2
a^2
−
(
c^2 +a^2 −b^2
2 a^2
) 2
=−
(c^2 +a^2 −b^2 )^2 − 4 c^2 a^2
4 a^4
=−
[(c+a)^2 −b^2 ][(c−a)^2 −b^2 ]
4 a^4
,
while
p 1 +
√
p^21 +q^21 =
c^2 +a^2 −b^2
2 a^2
+
c
a
=
(c+a)^2 −b^2
2 a^2
,
1 +
√
p^21 +q^21 +
√
( 1 −p 1 )^2 +q^21 = 1 +
c
a
+
b
a
=
a+b+c
a
,
so that
p 1 +
√
p^21 +q^21
1 +
√
p^21 +q^21 +
√
( 1 −p 1 )^2 +q^21
=
(c+a)^2 −b^2
2 a(a+b+c)
=
c+a−b
2 a
,
q^21
1 +
√
p^21 +q^21 +
√
( 1 −p 1 )^2 +q^21
=−
1
4 a^3
[(c+a)^2 −b^2 ][(c−a)^2 −b^2 ]
a+b+c
=−
1
4 a^3
(c+a−b)[(c−a)^2 −b^2 ].
The determinant for collinearity, on multiplying the middle column byq 1 ,is
∣∣
∣∣
∣∣
∣∣
∣
c^2 +a^2 −b^2
2 a^2
(c^2 +a^2 −b^2 )(a^2 +b^2 −c^2 )
4 a^41
c^2 +a^2 −b^2 + 2 a^2
6 a^2 −
[(c+a)^2 −b^2 ][(c−a)^2 −b^2 ]
12 a^41
c+a−b
2 a −
(c+a−b)[(c−a)^2 −b^2 ]
4 a^31