The triangle Δ N_1 N_2 N_3 formed by joining a set of three Neuberg centers (i.e., centers of the Neuberg circles) obtained from the edges of a given triangle Δ A_1 A_2 A_3. Similarly, a second set of three Neuberg circles with centers N_1^, , N_2^, , and N_3^, can be obtained from the main circles by reflection in their respective sides of the triangle, producing the second Neuberg triangle Δ N_1^, N_2^, N_3^, . The second Neuberg triangle has trilinear vertex matrix [a b c(a^2 + b^2 + c^2) | c(c^4 - a^2 c^2 - b^2 c^2 - 2a^2 b^2) | b(b^4 - a^2 b^2 - c^2 b^2 - 2a^2 c^2) c(c^4 - a^2 c^2 - b^2 c^2 - 2a^2 b^2) | a b c(a^2 + b^2 + c^2) | a(a^4 - b^2 a^2 - c^2 a^2 - 2b^2 c^2) b(b^4 - a^2 b^2 - c^2 b^2 - 2a^2 c^2) | a(a^4 - b^2 a^2 - c^2 a^2 - 2b^2 c^2) | a b c(a^2 + b^2 + c^2)].