Properties of the Triangle of Excentres

Properties of the Triangle of Excentres
Christopher Bradley
I3
To Y
B
1
I1
To X
O+ O K+ I
A
K
H
X
C
Article: CJB/218/2012
To Z
I2
Z

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
a 2 yz + b 2 zx + c 2 xy +(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, (1.2)
I2: (c + a)y + b(z + x) = 0, (1.3)
I3: (a + b)z + c(x + y) = 0. (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(a 3 + a 2 (b + c) – a(b + c) 2 – b 3 + b 2 c + bc 2 – c 3 ) (2.2)
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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 (2.3)
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.
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