The Yiu -circle of a reference triangle is the circle passing through vertex and the reflections of vertices and with respect to the opposite sides. The Yiu - and -circles are then analogously defined.
The -circle has center
(1)
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which can also be written
(2)
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(P. Moses, pers. comm., Jan. 31, 2005).
Its -radius is
(3)
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(4)
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where , , , and are Conway triangle notation, is the circumcenter, and is the orthocenter (P. Moses, pers. comm., Jan. 31, 2005).
The Yiu circles powers with respect to the vertices are
(5)
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(6)
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(7)
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(8)
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(9)
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The Yiu circles mutually intersect in a single point, which is therefore their radical center. It has center function
(10)
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which is Kimberling center (the inverse in the circumcircle of the Kosnita point ).
The Yiu circles do not have a radical circle.