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Yiu Circles


YiuCircles

The Yiu A-circle of a reference triangle DeltaABC is the circle passing through vertex A and the reflections of vertices B and C with respect to the opposite sides. The Yiu B- and C-circles are then analogously defined.

The A-circle has center

 (-a^8+4b^2a^6+4c^2a^6-6b^4a^4-6c^4a^4-5b^2c^2a^4+4b^6a^2+4c^6a^2-b^2c^4a^2-b^4c^2a^2-b^8-c^8+2b^2c^6-2b^4c^4+2b^6c^2):2a^2b^3c^3cos(A-C):2a^2b^3c^3cos(A-B)
(1)

which can also be written

 3a^2S_A^3-3S_AS_BS_C(3a^2+2S_A)-5S_A^2S_B^2-4S_B^2S_C^2-5S_C^2S_A^2:abc^2(S^2+S_CS_A):ab^2c(S^2+S_AS_B)
(2)

(P. Moses, pers. comm., Jan. 31, 2005).

Its A-radius is

R_A=(b^2c^2sqrt(S^2(OH^2+2S_A)+2S_AS_BS_C))/(S|b^2c^2-4S_A^2|)
(3)
=1/(|2cos(2A)+1|)sqrt((-a^6+3b^2a^4+3c^2a^4-3b^4a^2-3c^4a^2-3b^2c^2a^2+b^6+c^6-b^2c^4-b^4c^2)/(a^4-2b^2a^2-2c^2a^2+b^4+c^4-2b^2c^2)),
(4)

where S, S_A, S_B, and S_C are Conway triangle notation, O is the circumcenter, and H is the orthocenter (P. Moses, pers. comm., Jan. 31, 2005).

The Yiu circles powers with respect to the vertices are

p_A=0
(5)
p_B=(2accos(A-C))/(1+2cos(2A))
(6)
=2c^2(S^2+S_CS_A)
(7)
p_C=(2abcos(A-B))/(1+2cos(2A))
(8)
=2b^2(S^2+S_AS_B).
(9)
YiuCirclesRadicalCenter

The Yiu circles mutually intersect in a single point, which is therefore their radical center. It has center function

 alpha_(1157)=(a^6-3b^2a^4-3c^2a^4+3b^4a^2+3c^4a^2-b^2c^2a^2-b^6-c^6+b^2c^4+b^4c^2)/(cos(B-C)),
(10)

which is Kimberling center X_(1157) (the inverse in the circumcircle of the Kosnita point X_(54)).

The Yiu circles do not have a radical circle.


See also

Reflection, Yiu Circle, Yiu Triangle

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Cite this as:

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

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