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First Lemoine Circle

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FirstLemoineCircle

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

l=-(bc(b^2+c^2))/((a^2+b^2+c^2)^2),
(1)

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

R_L=1/2Rsecomega
(2)
=(abcsqrt(a^2b^2+b^2c^2+c^2a^2))/((a^2+b^2+c^2)sqrt((-a+b+c)(a-b+c)(a+b-c)(a+b+c))),
(3)

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

A_BP_B:P_BQ_C:Q_CA_C=c^2:a^2:b^2.
(4)

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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References

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." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X182.Lachlan, 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. https://mathworld.wolfram.com/FirstLemoineCircle.html

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