Cosine Circle


Draw antiparallels through the symmedian point K. 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 Q_2P_3, Q_3P_1, and Q_1P_2 are proportional to the cosines of the angles of DeltaA_1A_2A_3, 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 lambda=0. It has circle function


corresponding to Kimberling center X_(69). This gives it a center at the symmedian point K and a radius


where (2) also follows from the equation for Tucker circles


with lambda=0.

Kimberling centers X_(1666) and X_(1667) (the intersections with the Brocard axis) lie on the cosine circle.

Triangles DeltaP_1P_2P_3 and DeltaQ_2Q_3Q_1 are congruent, and symmetric with respect to the symmedian point. The sides of DeltaP_1P_2P_3 and DeltaQ_2Q_3Q_1 are to the sides of DeltaA_1A_2A_3 (P_1P_2 to A_3A_1, P_2P_3 to A_1A_2 and P_3P_1 to A_2A_3). The Miquel points of DeltaP_1P_2P_3 and DeltaQ_2Q_3Q_1 are the Brocard points.

See also

Brocard Circle, Brocard Points, Excosine Circle, Miquel Point, Second Brocard Circle, Taylor Circle, Tucker Circles

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Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, 1952.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., 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.

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Cosine Circle

Cite this as:

Weisstein, Eric W. "Cosine Circle." From MathWorld--A Wolfram Web Resource.

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