Nagel Line

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

			(2)
			(3)

The Nagel line is the radical line of the de Longchamps circle and Yff contact circle.