The Tucker circles are a generalization of the cosine circle and first Lemoine circle which can be viewed as a family of circles obtained by parallel displacing sides of the corresponding cosine or Lemoine hexagon. No matter how the segments are displaced, the Tucker hexagon will close, and the 12 vertices will be concyclic. The cosine circle and first Lemoine circle correspond to the special case where three sides of the Tucker hexagon concur.
Let three equal lines 
,
, and 
be drawn antiparallel
to the sides of a triangle so that two (say 
and 
) are on the same side of the third line as 
. Then 
is an isosceles trapezoid,
i.e., 
,
, and 
are parallel to the respective sides. The midpoints 
, 
, and 
of the antiparallels are on the respective symmedians and
divide them proportionally. If 
divides 
in the same ratio, 
, 
, 
are parallel to the radii 
, 
, and 
and equal. Since the antiparallels are perpendicular to
the symmedians, they form equal chords of a circle, called a Tucker circle, which
passes through the six given points and has center 
on the line 
(Honsberger 1995, pp. 92-94).
Defining
 |
(1)
|
then the radius of the Tucker circle is
 |
(2)
|
where 
is the Brocard angle and 
is the circumradius of the
reference triangle (Johnson 1929, p. 274).
Tucker circles can also be parametrized by a parametric angle 
. The Tucker circle with parametric angle has radius
 |
(3)
|
where 
is the Brocard angle and 
the circumradius of the reference triangle (Gallatly 1913, p. 134),
and trilinear center function
 |
(4)
|
Special named Tucker circles are summarized in the following table, where 
is the inradius, 
the semiperimeter of the
reference triangle, and 
and 
is Conway triangle
notation.
| Tucker circle |  |  |
| Apollonius circle |  |  |
| circumcircle | 0 | 1 |
| cosine circle |  | 0 |
| first Lemoine circle |  |  |
| Gallatly circle |  |  |
| Kenmotu circle |  |  |
| Taylor circle |  |  |
The Tucker circles are a coaxaloid system (Johnson 1929, p. 277).
The internal and external centers of similitudes of a Tucker circle with the nine-point circle lie on the Kiepert hyperbola (P. Moses, pers. comm., Jan. 3, 2005).
The two intersections of a Tucker circle with the Brocard axis have center functions
 |
(5)
|
and the inner and outer centers of similitude with the Brocard circle are given by
 |
(6)
|
where
 |
(7)
|
(pers. comm., P. Moses, Jan. 3, 2005).
