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Cyclocevian Conjugate


CyclocevianConjugate

Let P=alpha:beta:gamma be a point not on a sideline of a reference triangle DeltaABC. Let A^' be the point of intersection AP intersection BC, B^'=BP intersection AC, and C^'=CP intersection AB. The circle through A^'B^'C^' meets each sideline in two (not necessarily distinct) points A^', A^(''); B^', B^(''); and C^', C^(''). Then the lines AA^(''), BB^(''), and CC^('') concur in a point X known as the cyclocevian conjugate of P. The point has triangle center function

 alpha_X= 
 1/(a(gamma^2alpha^2+alpha^2beta^2-beta^2gamma^2)+2alphabetagamma(aalpha+bbeta+cgamma)cosA)

(Kimberling 1998, p. 226).

The Gergonne point Ge is its own cyclocevian conjugate.

The following table summarized cyclocevian conjugates for some triangle centers.


See also

Cyclocevian Triangle

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References

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Cyclocevian Conjugate

Cite this as:

Weisstein, Eric W. "Cyclocevian Conjugate." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CyclocevianConjugate.html

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