The Three Neuberg Circles
of a Triangle
by
Markus Heisss
Würzburg, Bavaria
2019/2024
Last update: June 3, 2024
The copying of the following graphics is allowed, but without changes.
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The three Neuberg circles of a given triangle
are always internally erected.
(see figure further below)
They are named: a-, b-, and c-Neuberg circle.
The c-Neuberg circle is the locus of vertex C
with base AB and a given Brocard angle ω.
The Brocard angle ω can be calculated as follows:
Example of the three Neuberg circles of a triangle:
Fig. 01: The three Neuberg circles of a triangle
(Note that the formulas for x and y contain the Brocard angle.)
You can copy the formulas for the calculation:
Delta=1/4*SQRT(2*(a*a*b*b+b*b*c*c+c*c*a*a)-(a^4+b^4+c^4))
cotOmega=(a^2+b^2+c^2)/(4*Delta)
x=1/2*SQRT(cotOmega^2-3)
y=1/2*cotOmega
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Further there is also a Neuberg triangle and the Neuberg circle:
Fig. 02: Neuberg triangle and Neuberg circle
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It is possible to construct six triangles directly or inverse similar to a given triangle.
If side AB is the base, than the other vertices lie all on the c-Neuberg circle.
See next graphic:
Fig. 03: The Neuberg circle and six similar triangles
(... from: Roger A. Johnson: 'Advanced Euclidean Geometry', p.289, Theorem 483)
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There is a simple relationship between the Neuberg circles
and the so-called McCay circles: The factor 3,
... as it can be see in the following graphic:
CMc = 3 GMc
(G is the centroid of triangle ABC, and Mc the midpoint of side AB.)
Fig. 04: c-Neuberg circle and c-McCay circle
And a further relationship:
Fig. 04: c-Neuberg circle with tangents and collinearities
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More about the Neuberg circles you can find here:
http://mathworld.wolfram.com/NeubergCircles.html
But much more interesting are the McCay circles!
The website: https://mccay-circles.jimdofree.com/
By the way: The circles were discovered by Joseph Neuberg.
More Information about his person you find here: