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Lucas Inner Triangle


LucasInnerTriangle

The points of tangency of the Lucas inner circle with the Lucas circles are the inverses of the vertices A, B, and C in the Lucas circles radical circle. These form the Lucas inner triangle DeltaI_AI_BI_C, a term coined here for the first time.

The Lucas inner triangle has trilinear vertex matrix

 [a(4S_A+3S) 2b(2S_B+S) 2c(2S_C+S); 2a(2S_A+S) b(4S_B+3S) 2c(2S_C+S); 2a(2S_A+S) 2b(2S_B+S) c(4S_C+3S)],

where S_A, S_B, and S_C is Conway triangle notation (P. Moses, pers. comm., Jan. 13, 2005).

The area is

 Delta_L=(a^2b^2c^2(a^2+b^2+c^2+7Delta))/((3a^2+2b^2+2c^2+16Delta)(2a^2+3b^2+2c^2+16Delta)(2a^2+2b^2+3c^2+16Delta)),

where Delta is the area of the reference triangle.

The following table gives the centers of the Lucas inner triangle in terms of the centers of the reference triangle that correspond to Kimberling centers X_n.

The following table summarized the points at which the Lucas inner triangle is perspective with various other triangles.


See also

Lucas Circles, Lucas Inner Circle

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Cite this as:

Weisstein, Eric W. "Lucas Inner Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LucasInnerTriangle.html

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