Yiu Circles -- from Wolfram MathWorld

Yiu Circles

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

(-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 -radius is

			(3)
			(4)

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)
			(6)
			(7)
			(8)
			(9)

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 (the inverse in the circumcircle of the Kosnita point ).

The Yiu circles do not have a radical circle.

Yiu Circles

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