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, three reflected Neuberg
circles with centers 
, 
, and 
can be obtained from the main circles by reflection in
their respective sides of the triangle, producing a reflected Neuberg triangle 
(right figure).
The Neuberg triangle has trilinear vertex matrix
![[abc(a^2+b^2+c^2) c(-a^4+c^2a^2-b^4+b^2c^2) b(-a^4+b^2a^2-c^4+b^2c^2); c(-a^4+c^2a^2-b^4+b^2c^2) abc(a^2+b^2+c^2) a(-b^4+a^2b^2-c^4+a^2c^2); b(-a^4+b^2a^2-c^4+b^2c^2) a(-b^4+a^2b^2-c^4+a^2c^2) abc(a^2+b^2+c^2)]](/images/equations/FirstNeubergTriangle/NumberedEquation1.svg) |
(1)
|
The Neuberg triangle has side lengths
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
and area
 |  |  |
(5)
|
 |  |  |
(6)
|
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 
,
,
and 
are concurrent at the Tarry point 
(Gallatly 1913; Johnson 1929, p. 288; left figure) which
has triangle center function
 |
(7)
|
where 
is the Brocard angle, and is Kimberling's center
.
The circumcircle of the first Neuberg triangle is the first Neuberg circle.
