Lemoine Circle
Also called the Triplicate-Ratio Circle. Draw lines through the Lemoine Point 
and parallel to the sidesof the triangle. The points where the parallel lines intersect the sides then lie on a Circle known as the Lemoinecircle. This circle has center at the Midpoint of 
, where 
is the Circumcenter. The circle has radius
where
is the Circumradius, 
is the radius of the Cosine Circle, and 
is the BrocardAngle. The 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 Cosine Circle is sometimes called the second Lemoine circle.
See also Cosine Circle, Lemoine Line, Lemoine Point, Tucker Circles
References
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.
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