The Lucas central triangle (a term coined here for the first time) is the triangle 
formed by the centers of the Lucas circles of a
given reference triangle 
.
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)],](/images/equations/LucasCentralTriangle/NumberedEquation1.svg) |
(1)
|
where 
,
,
,
and 
is Conway triangle notation.
The Lucas central triangle has side lengths
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
Its area is given by
 |
(5)
|
where 
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 
with 
.
 | center of Lucas central triangle |  | center of reference triangle |
 | incenter |  | isogonal conjugate of  |
 | isoperimetric point |  | circumcenter |
 | first Eppstein point |  | Kenmotu point |
The Cevian triangles of Kimberling centers 
for 
, 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]](/images/equations/LucasCentralTriangle/NumberedEquation3.svg) |
(6)
|
is perspective to the Lucas central triangle (P. Moses, pers. comm., Feb. 3 2005). The anticevian triangles of 
for 
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.
| triangle | perspector |
| Lucas inner triangle |  |
| Lucas tangents triangle |  |
| symmedial triangle |  |
