The Nagel line is the term proposed for the first time in this work for the line on which the incenter 
, triangle centroid 
, Spieker
center Sp, and Nagel point Na lie.
Because Kimberling centers 
and 
both lie on this line, it is denoted
and is the first line in Kimberling's
enumeration of central lines containing at least three collinear centers (Kimberling
1998, p. 128).
The Kimberling centers 
lying on the line include 
(incenter 
), 2 (triangle centroid 
), 8 (Nagel
point Na), 10 (Spieker center Sp),
42, 43, 78, 145, 200, 239, 306, 386, 387, 498, 499, 519, 551, 612, 614, 869, 899,
936, 938, 975, 976, 978, 995, 997, 1026, 1103, 1125, 1149, 1189, 1193, 1198, 1201,
1210, 1644, 1647, 1698, 1714, 1722, 1737, 1961, 1998, 1999, 2000, 2057, 2340, 2398,
2534, 2535, 2664, 2999, 3006, 3008, 3009, 3011, and 3017.
The Nagel line is central line 
, so its trilinear equation is
 |
(1) |
The Nagel line satisfies the remarkable property of being its own complement, and therefore also its own anticomplement.
The incenter 
, Spieker center Sp,
Nagel point Na, and triangle
centroid 
satisfy the distance relations
The Nagel line is the radical line of the de Longchamps circle and Yff contact circle.
See also
Incenter, Nagel Point, Spieker Center, Triangle Centroid
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References
Honsberger, R. "The Nagel Point 
and the Spieker Circle." ยง1.4 in Episodes
in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math.
Assoc. Amer., pp. 5-13, 1995.Kimberling, C. "Triangle Centers
and Central Triangles." Congr. Numer. 129, 1-295, 1998.
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Cite this as:
Weisstein, Eric W. "Nagel Line." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NagelLine.html