Download Properties of the Triangle of Excentres

Article: CJB/218/2012
Properties of the Triangle of Excentres
Christopher Bradley
X
To X
I3
A
I2
O+ O K+I
K
H
To Y
C
B
To Z
Z
I1
1
Abstract: In the triangle of excentres the orthocentre is the incentre I of ABC, the Symmedian
point K+ is the Mittenpunkt of ABCand the nine-point centre is the circumcentre O of ABC. The
circumcentre O+ lies on the line OI and is such that O+O = OI. These properties are proved as
also the fact that KI passes through K+.
1. Circle I1I2I3 and its tangents
The co-ordinates of the excentres are I1(– a, b, c), I2(a, – b, c), I3(a, b, – c). It is easy to check that
the equation of the circle I1I2I3 is
a2yz + b2zx + c2xy +(x + y + z)(bcx + cay + abz) = 0.
(1.1)
The tangents to this circle at the points I1, I2, I3 have equations
I1:
(b + c)x + a(y + z) = 0,
I2:
(c + a)y + b(z + x) = 0,
I3:
(a + b)z + c(x + y) = 0.
(1.2)
(1.3)
(1.4)
Tangents at I2, I3 meet at X with co-ordinates X(– a(a + b + c), b(a + b – c), c(a – b + c)).
Tangents at I3, I1 meet at Y with co-ordinates Y(a(a + b – c), – b(a + b + c),c(b + c – a)).
Tangents at I1, I2 meet at Z with co-ordinates Z(a(c + a – b), b(b + c – a), – c(a + b + c)).
2. The symmedian point K+
The symmedian point K+ of triangle I1I2I3 is the point of concurrence of lines XI1, YI2, ZI3
whose equations are respectively
(c – b)x = a(y – z), (a – c)y = b(z – x), (b – a)z = c(x – y).
(2.1)
The co-ordinates of K+ are therefore (a(b + c – a), b(c + a – b), c(a + b – c)). The co-ordinates
will be recognized as those of the Mittenpunkt Mi, X9 in Kimberling’s list of Triangle Centres.
It is a well known fact that in triangle I1I2I3 that II1 is perpendicular to I2I3 etc. Hence the incentre
I is the orthocentre H+ of triangle I1I2I3.
From the usual formulae for the centre of a conic we find the co-ordinates of O+ to be (x, y, z),
where
x = a(a3 + a2(b + c) – a(b + c)2 – b3 + b2c + bc2 – c3)
(2.2)
2
with y, z following by cyclic change of a, b, c. It may now be checked that this is the point on IO
such that O+O = OI and hence the nine-point centre of triangle I1I2I3 is the circumcentre O of
triangle ABC.
The equation of the line IK is
bc(c – b)x + ca(a – c)y + ab(b – a)z = 0
and it may be checked that K+ lies on this line.
Finally, after some heavy algebra, it may be checked that K+ lies on the conic I1I2I3OH.
Flat 4,
Terrill Court,
12-14, Apsley Road,
BRISTOL BS8 2SP.
3
(2.3)