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).

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.