Draw lines ,
,
and
through the symmedian point and parallel to the sides of the triangle . The points where the parallel lines intersect
the sides of
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

(1)

corresponding to Kimberling center , which is the complement of the symmedian point. It
has center at the Brocard midpoint , i.e., the midpoint of , where is the circumcenter and is the symmedian
point , and radius

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

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

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

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

(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. Referenced on Wolfram|Alpha 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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