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McCay Circles


McCayCircles

The three circumcircles through the triangle centroid G of a given triangle DeltaA_1A_2A_3 and the pairs of the vertices of the second Brocard triangle are called the McCay circles (Johnson 1929, p. 306).

The circumcircle of their centers (i.e., of the second Brocard triangle) is therefore the Brocard circle.

The A-McCay circle has center function

 alpha:beta:gamma=2bccosA:ab:ac.

and radius,

 R_A=1/6asqrt(cot^2omega-3),

1/3 that of the Neuberg circle, where omega is the Brocard angle (Johnson 1929, p. 307).

McCay circle

If the polygon vertex A_1 of a triangle describes a Neuberg circle N_1, then its triangle centroid G describes one of the McCay circles (Johnson 1929, p. 290). In the above figure, the inner triangle is the second Brocard triangle of DeltaA_1A_2A_3, whose two indicated edges are concyclic with G on the McCay circle.


See also

Brocard Circle, Circle, Concurrent, McCay Circles Radical Circle, Neuberg Circles, Second Brocard Triangle, Triangle Centroid

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References

Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, pp. 83-84 and 128-129, 1971.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 290 and 306-307, 1929.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 145 and 222, 1893.M'Cay, W. S. "On Three Circles Related to a Triangle." Trans. Roy. Irish Acad. 28, 453-470, 1885.

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McCay Circles

Cite this as:

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

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