Lemoine Line
The Lemoine line, also called the Lemoine Axis, is the perspectivity axis of a Triangle and itsTangential Triangle, and also the Trilinear Polar of the Centroid of thetriangle vertices. It is also the Polar of 
with regard to its Circumcircle, and isPerpendicular to the Brocard Axis.
The centers of the Apollonius Circles 
, 
, and 
are Collinear on the Lemoine Line. This line is Perpendicular to the Brocard Axis 
and is the Radical Axis of the Circumcircle and the Brocard Circle. It has equation
in terms of Trilinear Coordinates (Oldknow 1996).See also Apollonius Circles, Brocard Axis, Centroid (Triangle),Circumcircle, Collinear, Lemoine Circle, Lemoine Point,Polar, Radical Axis,Tangential Triangle, Trilinear Polar
References
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 295, 1929. Oldknow, A. ``The Euler-Gergonne-Soddy Triangle of a Triangle.'' Amer. Math. Monthly 103, 319-329, 1996.
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