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First Neuberg Triangle


NeubergTriangles

The triangle DeltaN_1N_2N_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 DeltaA_1A_2A_3 (left figure). Similarly, three reflected 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 a reflected Neuberg triangle DeltaN_1^'N_2^'N_3^' (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)]
(1)

The Neuberg triangle has side lengths

a^'=sqrt((b^6-a^2b^2c^2+b^4c^2+b^2c^4+c^6)/((-a+b+c)(a-b+c)(a+b-c)(a+b+c)))
(2)
b^'=sqrt((a^6+a^4c^2-a^2b^2c^2+a^2c^4+c^6)/((-a+b+c)(a-b+c)(a+b-c)(a+b+c)))
(3)
c^'=sqrt((a^6+a^4b^2+a^2b^4+b^6-a^2b^2c^2)/((-a+b+c)(a-b+c)(a+b-c)(a+b+c)))
(4)

and area

Delta^'=Delta/(4sin^2omega)
(5)
=(a^2b^2+a^2c^2+b^2c^2)/(4sqrt((-a+b+c)(a-b+c)(a+b-c)(a+b+c))).
(6)
NeubergTriangleCentroids

The triangle centroid G_N of DeltaN_1N_2N_3 is coincident with the triangle centroid G_A of DeltaA_1A_2A_3 (Gallatly 1913; Johnson 1929, p. 288; left figure). Similarly, the centroids of DeltaA_1A_2A_3 and DeltaN_1^'N_2^'N_3^' also coincide (right figure).

NeubergTriangleLines

The lines A_1N_1, A_2N_2, and A_3N_3 are concurrent at the Tarry point T (Gallatly 1913; Johnson 1929, p. 288; left figure) which has triangle center function

 alpha_(98)=sec(A+omega),
(7)

where omega is the Brocard angle, and is Kimberling's center X_(98).

The circumcircle of the first Neuberg triangle is the first Neuberg circle.


See also

First Neuberg Circle, Neuberg Center, Neuberg Circles, Second Neuberg Triangle, Tarry Point

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References

Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, 1913.Grinberg, D. "Neuberg triangles, X(262) - Two Tarry points? Two 3rd Brocard points? [typos corrected]." geometry-college@mathforum.org mailing list. 12 Jan 2003.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.

Referenced on Wolfram|Alpha

First Neuberg Triangle

Cite this as:

Weisstein, Eric W. "First Neuberg Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FirstNeubergTriangle.html

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