Yff Central Triangle


Let three isoscelizers I_(AC)I_(AB), I_(BA)I_(BC), and I_(CA)I_(CB) be constructed on a triangle DeltaABC, one for each side. This makes all of the inner triangles similar to each other. However, there is a unique set of three isoscelizers for which the four interior triangles DeltaA^'I_(BC)I_(CB), DeltaI_(AC)B^'I_(CA), DeltaI_(AB)I_(BA)C^', and DeltaA^'B^'C^' are congruent. The innermost triangle DeltaA^'B^'C^' is called the Yff central triangle (Kimberling 1998, pp. 94-95).

It has trilinear vertex matrix

 [yz z(x+z) y(x+y); z(y+z) zx x(y+x); y(z+y) x(z+x) xy],

where x=cos(A/2), y=cos(B/2), and z=cos(C/2) (Kimberling 1998, p. 172; typo corrected).


The original triangle DeltaABC is the extangents triangle of the Yff central triangle DeltaA^'B^'C^'.

The circumcircle of the Yff central circle is the Yff central circle.

The following table gives the centers of the Yff central triangle in terms of the centers of the reference triangle for Kimberling centers X_n with n<=100.

See also

Extangents Triangle, Isoscelizer, Yff Center of Congruence, Yff Central Circle

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

Referenced on Wolfram|Alpha

Yff Central Triangle

Cite this as:

Weisstein, Eric W. "Yff Central Triangle." From MathWorld--A Wolfram Web Resource.

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