There are two nonintersecting circles that are tangent to all three Lucas circles. (These are therefore the Soddy circles of the Lucas central triangle.) The inner one, dubbed the Lucas inner circle here for the first time, is the inverse of the circumcircle in the Lucas circles radical circle (P. Moses, pers. comm., Jan. 3, 2005).
Its center has triangle center function
 |
(1)
|
where 
is the area of the reference triangle, which
lies on the Brocard axis, and its radius is
 |  | ![(abc)/(4[(a^2+b^2+c^2)+7Delta])](/images/equations/LucasInnerCircle/Inline4.svg) |
(2)
|
 |  |  |
(3)
|
where 
is the Brocard angle (P. Moses, pers. comm.,
Jan. 3, 2005).
It has circle function
 |
(4)
|
corresponding to the triangle centroid 
, which is Kimberling
center 
.
It is orthogonal to the Parry circle.
The circumcircle, Lucas circles radical circle, Lucas inner circle, and Brocard circle share the Lemoine axis as their radical line and hence are part of the Schoute coaxal system (P. Moses, pers. comm., Jan. 3, 2005).
