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.

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Carr, G. S. Art. 4754b in Synopsis of Elementary Results in Pure Mathematics, 2nd ed., 2 vols. New York: Chelsea, 1970.

Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 66, 1971.

Ehrmann, J.-P. and van Lamoen, F. M. "The Stammler Circles." Forum Geom. 2, 151-161, 2002. http://forumgeom.fau.edu/FG2002volume2/FG200219index.html.

Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 117, 1913.

Honsberger, R. "The Lemoine Circles." §9.2 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 88-89, 1995.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 271-273, 1929.

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Lachlan, R. "The Cosine Circle." §129-130 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, p. 75, 1893.

CITE THIS AS:

Weisstein, Eric W. "Cosine Circle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CosineCircle.html