Tucker Circles

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).