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Second Brocard Circle


SecondBrocardCircle

The second Brocard circle is the circle having center at the circumcenter O of the reference triangle and radius

R_B=sqrt(1-4sin^2omega)R
(1)
=abcsqrt((a^4+b^4+c^4-a^2b^2-a^2c^2-b^2c^2)/((-a+b+c)(a-b+c)(a+b-c)(a+b+c)(a^2b^2+a^2c^2+b^2+c^2))),
(2)

where R is the circumradius. It has circle function

 l=-(a^2bc)/(a^2b^2+a^2c^2+b^2c^2)
(3)

which corresponds to the symmedian point K.

The (first) Brocard circle and second Brocard circle meet in the first and second Brocard points.

The second Brocard circle also passes through the points X_i for i=1670 and 1671.


See also

Brocard Circle, Cosine Circle

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References

Kimberling, C. "Clark Kimberling's Encyclopedia of Triangle Centers--ETC." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X2445.

Referenced on Wolfram|Alpha

Second Brocard Circle

Cite this as:

Weisstein, Eric W. "Second Brocard Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SecondBrocardCircle.html

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