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Neuberg Circles


Neuberg circle

The Neuberg A_1-circle is the locus of the polygon vertex A_1 of a triangle on a given base A_2A_3 and with a given Brocard angle omega. From the center N_1, the base A_2A_3 subtends the angle 2omega. The same procedure can be repeated for the other two sides of a triangle resulting in a total of three Neuberg circles. 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.

The equation of the A_1-circle can be found by taking the base as (0, 0), (a_1, 0) and solving

x^2+y^2=a_3^2
(1)
(x-a_1)^2+y^2=a_2^2
(2)

while eliminating a_2 and a_3 using

 cotomega=(a_1^2+a_2^2+a_3^3)/(4Delta),
(3)

where Delta is the area of the triangle DeltaA_1A_2A_3. Solving for x gives

 x=1/2(a_1+/-sqrt(+/-4a_1ycotomega-4y^2-3a_1^2)),
(4)

and squaring and completing the square results in

 (x-1/2a_1)^2+(y+/-1/2a_1cotomega)^2=1/4a_1^2(cot^2omega-3).
(5)

Therefore, the Neuberg circle N_1 on this edge has center

 N_1=(1/2a_1,+/-1/2a_1cotomega)
(6)

and radius

 r=1/2a_1sqrt(cot^2omega-3).
(7)

The centers of the Neuberg circles are called Neuberg centers, and the triangles determined by the Neuberg centers are called the first and second Neuberg triangles.

No Kimberling centers lie on any of the Neuberg circles or Neuberg reflected circles.

The circle parameters of the Neuberg circles are given by

(l_A,m_A,n_A)=(0,a/c,a/b)
(8)
(l_B,m_B,n_B)=(b/c,0,b/a)
(9)
(l_C,m_C,n_C)=(c/b,c/a,0).
(10)
NeubergCircles

The first Neuberg triangle DeltaN_1N_2N_3 (left figure) and reflected first Neuberg triangle DeltaN_1^'N_2^'N_3^' (right figure) are illustrated above.

On one side of a given line taken as a base, it is possible to construct six triangles directly or inversely similar to a given scalene triangle, and the vertices of these triangles lie on their common Neuberg circles (Johnson 1929, p. 289).


See also

Brocard Angle, First Neuberg Triangle, McCay Circles, Neuberg Center, Neuberg Circles Radical Circle, Neuberg Cubic, Second Neuberg Triangle

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References

Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 79-80, 1971.Emmerich, A. Die Brocardschen Gebilde und ihre Beziehungen zu den verwandten merkwürdigen Punkten und Kreisen des Dreiecks. Berlin: Reimer, 1891.Gallatly, W. "Neuberg Circles." §135 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 97, 1913.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 287-290, 1929.PandD Software. "Neuberg-Cirkels." http://www.pandd.demon.nl/lemoine/neuberg.htm.

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Neuberg Circles

Cite this as:

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

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