Nail point
The Nagel point , named after the German mathematician Christian Heinrich von Nagel (1803–1882), who showed the existence of this point in 1835/36, is one of the special points of a triangle . For a given triangle ABC, consider points D, E and F, where the circles touch the sides of the triangle. If you connect these points of contact with the opposite corners of the triangle (i.e. with A, B or C), these connecting lines intersect at a point N. This is called the nail point of the triangle.
properties
- If, in addition to the Nagel point N of the triangle ABC, one also considers the inscribed center point I and the center of gravity S, then the points N, S and I lie on a straight line, the Nagel line , and the following applies , the center of gravity S between the points N. and I lies. In this property, the nail straight line has an analogy to Euler's straight line .
- The Spieker point is the center point of the connecting line between the nail point and the inscribed center point and is therefore also on the straight line of the nail.
- The nail point and the Gergonne point are isotomically conjugated .
Coordinates
| Nail point ( ) | |
|---|---|
| Trilinear coordinates | |
| Barycentric coordinates | |
literature
- Peter Baptist: Historical Notes on Gergonne and Nagel Points. In: Sudhoffs Archiv 71, 1987, 2, pp. 230-233
- Roger A. Johnson : Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 225-229 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry ).
- Edwin Kozniewski, Renata A. Gorska: Gergonne and Nagel Points for Simplices in the n-Dimensional Space . Journal for Geometry and Graphics, Vol. 4, 2000, No. 2, pp. 119-127
- Victor Thébault : Nagel Point in the Tetrahedron . The American Mathematical Monthly, Volume 54, No. 5 (May, 1947), pp. 275-276 ( JSTOR 2305352 )