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)
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corresponding to Kimberling center . This gives it a center at the symmedian point and a radius
(2)
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(3)
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where (2) also follows from the equation for Tucker circles
(4)
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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.