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Lucas Central Triangle


LucasCentralTriangle

The Lucas central triangle (a term coined here for the first time) is the triangle DeltaL_AL_BL_C formed by the centers of the Lucas circles of a given reference triangle DeltaABC.

It has trilinear vertex matrix

 [a(2S+S_A) bS_B cS_C; aS_A b(2S+S_B) cS_C; aS_A bS_B c(2S+S_C)],
(1)

where S, S_A, S_B, and S_C is Conway triangle notation.

The Lucas central triangle has side lengths

a^'=(2R(abc+b^2R+c^2R))/((ac+2bR)(ab+2cR))a
(2)
b^'=(2R(abc+a^2R+c^2R))/((bc+2aR)(ab+2cR))b
(3)
c^'=(2R(abc+a^2R+b^2R))/((bc+2aR)(ac+2bR))c.
(4)

Its area is given by

 Delta=(abcR^2sqrt(3a^2b^2c^2+4abc(a^2+b^2+c^2)R+4(a^2b^2+a^2c^2+b^2c^2)R^2))/((bc+2aR)(ac+2bR)(ab+2cR)),
(5)

where R is the circumradius of the reference triangle.

The circumcircle of the Lucas central triangle is the Lucas central circle.

The following table gives the centers of the Lucas central triangle in terms of the centers of the reference triangle for Kimberling centers X_n with n<=1000.

The Cevian triangles of Kimberling centers X_i for i=3, 6, 371, and 588 are perspective to the Lucas central triangle. In fact, the Cevian triangle of any point lying on the trilinear cubic

 sum_(cyclic)a^2S_A[c(b^2+2S)beta^2gamma-b(c^2+2S)betagamma^2]
(6)

is perspective to the Lucas central triangle (P. Moses, pers. comm., Feb. 3 2005). The anticevian triangles of X_i for i=3 and 6 are also perspective to the Lucas central triangle (P. Moses, pers. comm., Jan. 21, 2005). The following table summarizes some perspectors of the Lucas central triangle and other named triangles.


See also

Lucas Central Circle, Lucas Circles, Lucas Circles Radical Circle, Lucas Tangents Triangle

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Cite this as:

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

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