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 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).
![[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_400.svg)