Geometry with Trigonometry

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

∣∣

∣∣

∣∣

∣∣

∣

,