First Lemoine Circle


Draw lines P_AQ_A, P_BQ_B, and P_CQ_C through the symmedian point K and parallel to the sides of the triangle DeltaABC. The points where the parallel lines intersect the sides of DeltaABC then lie on a circle known as the first Lemoine circle, or sometimes the triplicate-ratio circle (Tucker 1883; Kimberling 1998, p. 233).

This circle has circle function


corresponding to Kimberling center X_(141), which is the complement of the symmedian point. It has center at the Brocard midpoint X_(182), i.e., the midpoint of OK, where O is the circumcenter and K is the symmedian point, and radius


where R is the circumradius, r is the inradius, and omega is the Brocard angle of the original triangle (Johnson 1929, p. 274).

Kimberling centers X_(1662) and X_(1664) (the intersections with the Brocard axis) lie on the first Lemoine circle.

The first Lemoine circle and Brocard circle are concentric, and the triangles DeltaQ_AP_CK, DeltaKQ_CP_B, and DeltaP_AKQ_B are similar to DeltaACB (Tucker 1883).

The first Lemoine circle divides any side into segments proportional to the squares of the sides


Furthermore, the chords cut from the sides by the Lemoine circle are proportional to the squares of the sides.

The first Lemoine circle is a special case of a Tucker circle.

See also

Cosine Circle, Lemoine Hexagon, Lemoine Axis, Symmedian Point, Taylor Circle, Third Lemoine Circle, Tucker Circles

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Casey, J. "On the Equations and Properties--(1) of the System of Circles Touching Three Circles in a Plane; (2) of the System of Spheres Touching Four Spheres in Space; (3) of the System of Circles Touching Three Circles on a Sphere; (4) of the System of Conics Inscribed to a Conic, and Touching Three Inscribed Conics in a Plane." Proc. Roy. Irish Acad. 9, 396-423, 1864-1866.Casey, J. "Lemoine's, Tucker's, and Taylor's Circle." Supp. Ch. §3 in A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 179-189, 1888.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 70, 1971.Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 116, 1913.Honsberger, R. "The Lemoine Circles" and "The First Lemoine Circle." §9.2 and 9.5 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 88-89 and 94-95, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 273-275, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Encyclopedia of Triangle Centers: X(182)=Midpoint of Brocard Diameter.", R. "The Lemoine Circle." §131-132 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 76-77, 1893.Lemoine. Assoc. Français pour l'avancement des Sci. 1873.Tucker, R. "The 'Triplicate Ratio' Circle." Quart. J. Pure Appl. Math. 19, 342-348, 1883.

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First Lemoine Circle

Cite this as:

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

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