Draw antiparallels through the symmedian point 
.
The points where these lines intersect the sides
then lie on a circle, known as the cosine circle (or sometimes
the second Lemoine circle). The chords 
, 
, and 
are proportional to the cosines
of the angles of 
, giving the circle its name. In fact, there are
infinitely many circles that cut the side line chords in the same proportions. The
centers of these circles lie on the Stammler hyperbola
(Ehrmann and van Lamoen 2002).
The cosine circle is a special case of a Tucker circle with 
.
It has circle function
 |
(1)
|
corresponding to Kimberling center 
. This gives it a center at the symmedian
point 
and a radius
 |  |  |
(2)
|
 |  |  |
(3)
|
where (2) also follows from the equation for Tucker circles
 |
(4)
|
with 
.
Kimberling centers 
and 
(the intersections with the Brocard axis) lie on
the cosine circle.
Triangles 
and 
are congruent, and symmetric with respect to the symmedian
point. The sides of 
and 
are to the sides of 
(
to 
, 
to 
and 
to 
). The Miquel points of
and 
are the Brocard
points.
