Lemoine Axis -- from Wolfram MathWorld
- ️Weisstein, Eric W.
The Lemoine axis is the perspectrix of a reference triangle and its tangential triangle,
and also the trilinear polar of the symmedian
point 
of the reference triangle. It is also the polar of 
with regard to the circumcircle,
and is perpendicular to the Brocard
axis.
The centers of the Apollonius circles are collinear on the Lemoine axis. This line is perpendicular
to the Brocard axis 
and is the radical line of
the circumcircle and the Brocard
circle.
It is central line 
(Kimberling 1998, p. 150) and has trilinear equation
(Oldknow 1996). It passes through Kimberling centers 
for 
(Schoute center), 237, 351 (center of the Parry
circle), 512, 647, 649, 663, 665, 667, 669, 887, 890, 902, 1055, 1495, 1960,
2223, 2488, 2502, 2509, 2978, 3005, 3009, 3010, and 3016.
The Lemoine axis is the radical line of the coaxal system (Brocard circle, circumcircle, Lucas circles radical circle, Lucas inner circle), which includes the circumcircle and Brocard circle as special cases (Casey 1888, p. 177; Kimberling 1998, p. 150).
See also
Apollonius Circle, Brocard Axis, Circumcircle, Collinear, First Lemoine Circle, Symmedian Point, Polar, Radical Line, Symmedian, Tangential Triangle, Triangle Centroid, Trilinear Polar
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References
Casey, J. 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., 1888.Gallatly, W. "The Lemoine Axis." §128 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 92, 1913.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 295, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319-329, 1996.
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Cite this as:
Weisstein, Eric W. "Lemoine Axis." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LemoineAxis.html