The triangle 
formed by joining a set of three Neuberg
centers (i.e., centers of the Neuberg circles)
obtained from the edges of a given triangle 
(left figure). Similarly, a second set of three
Neuberg circles with centers 
, 
, and 
can be obtained from the main circles by reflection in
their respective sides of the triangle, producing the second Neuberg triangle 
(right figure).
The second Neuberg triangle has trilinear vertex matrix
![[abc(a^2+b^2+c^2) c(c^4-a^2c^2-b^2c^2-2a^2b^2) b(b^4-a^2b^2-c^2b^2-2a^2c^2); c(c^4-a^2c^2-b^2c^2-2a^2b^2) abc(a^2+b^2+c^2) a(a^4-b^2a^2-c^2a^2-2b^2c^2); b(b^4-a^2b^2-c^2b^2-2a^2c^2) a(a^4-b^2a^2-c^2a^2-2b^2c^2) abc(a^2+b^2+c^2)].](/images/equations/SecondNeubergTriangle/NumberedEquation1.svg) |
(1)
|
The triangle centroid 
of 
is coincident with the triangle
centroid 
of 
(Gallatly 1913; Johnson 1929, p. 288; left figure). Similarly, the centroids
of 
and 
also coincide (right figure).
The lines 
,
,
concur at a point having equivalent triangle
center functions
 |  |  |
(2)
|
 |  |  |
(3)
|
which is Kimberling center 
(right figure; Grinberg 2003).
