BCI Triangle


The BCI triangle DeltaA^'B^'C^' of a triangle DeltaABC with incenter I is defined by letting A^' be the center of the incircle of DeltaBCI, and similarly defining B^' and C^'.


The triangles DeltaABC and DeltaA^'B^'C^' are in perspective, the perspector being the first de Villiers point, which is Kimberling center X_(1127).

The BCI triangle has trilinear vertex matrix

 [1 1+2cos(1/2C) 1+2cos(1/2B); 1+2cos(1/2C) 1 1+2cos(1/2A); 1+2cos(1/2B) 1+2cos(1/2A) 1].

The following table gives some centers of the BCI triangle in terms of the centers of the reference triangle that correspond to Kimberling centers X_n for n<=1000.

X_ncenter of BCI triangleX_ncenter of reference triangle
X_(372)(X_3,X_6)-harmonic conjugate of X_(371)X_(483)radical center of the Malfatti circles
X_(486)inner Vecten pointX_1incenter

See also

de Villiers Points

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

Referenced on Wolfram|Alpha

BCI Triangle

Cite this as:

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

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