Also called the Triplicate-Ratio Circle. Draw lines through the Lemoine Point and parallel to the sides
of the triangle. The points where the parallel lines intersect the sides then lie on a Circle known as the Lemoine
circle. 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 Brocard Angle. 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.

**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.

© 1996-9

1999-05-26