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
(1)
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The Neuberg triangle has side lengths
(2)
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(3)
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(4)
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and area
(5)
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(6)
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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)
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where is the Brocard angle, and is Kimberling's center .
The circumcircle of the first Neuberg triangle is the first Neuberg circle.