Consider a reference triangle 
and externally inscribe a square on the side 
. Now join the new vertices 
and 
of this square with the vertex 
, marking the points of intersection 
and 
. Next, draw lines perpendicular to the side 
through each of 
and 
. These points cross the sides 
and 
at 
and 
, respectively, resulting in an inscribed square
. The circumcircle through 
, 
,
and 
is then known as the Lucas 
-circles (Panakis 1973, p. 458; Yiu and Hatzipolakis 2001),
and repeating the process for other sides gives the corresponding 
- and 
-circles.
The Lucas 
-circle
has the beautiful trilinear center
 |
where 
is the area of the reference triangle 
is the circumradius
of the reference triangle, and radius
 |
(Yiu and Hatzipolakis 2001).
The Lucas circles are pairwise tangent, although this fact seems to have been noted first only by Yiu and Hatzipolakis (2001).
There are two nonintersecting circles that are tangent to all three Lucas circles (these are the Soddy circles of the Lucas central triangle). The outer tangent circle is the circumcircle of the reference triangle, while the inner is the Lucas inner circle, which is the inverse of the circumcircle in the Lucas circles radical circle (P. Moses, pers. comm., Jan. 3, 2005).
There are also three circles analogous to the Lucas circles obtained when the original square is escribed instead of inscribed.
