# Nagel Line: What Is It About? A Mathematical Droodle

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Explanation

The applet may suggest the following interesting fact:

In any triangle ABC, four points, the incenter (I), the centroid (G), the Spieker center (S) and the Nagel point (M) are collinear. Moreover,

IS = SM,
IG = 2·GS,
MG = 2·IG.

Note that the situation is very much analogous to the collinearity of several notable points on the Euler line, on which four points, the circumcenter (O), the centroid (G), the 9-point center (N) and the orthocenter (H) are positioned in exactly the same ratios.

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The Spieker Center of a triangle is, by definition, the incenter of its medial triangle. It is named after the 19th century German geometer Theodor Spieker. It serves as the intersection of the cleavers of the reference triangle.

The key observation here is that the the centroid serves as the center of homothety, or dilation, with coefficient -1/2. If D, E, F are the midpoints of sides BC, AC, and AB, respectively, then D, E, F are the images of the vertices A, B, and C under this homothety. Homothety, as a similarity transformation, preserves relative locations of the corresponding points. In particular, the incenter of ΔABC maps on the incenter of ΔDEF, i.e. on its own Spieker center. Furthermore, the specific homothety with center at G and coefficient -1/2 reflects a point in G and shrinks it distance to G by half. Therefore, the points I, G, S are indeed collinear and IG = 2·GS. Similarly, if T denotes the Nagel point of ΔDEF, then the points M, G and T are collinear and MG = 2·GT. However, it is known that, quite conveniently, the Nagel point of ΔDEF coincides with the incenter of the base ΔABC. This gives MG = 2·IG. An additional identity, IS = SM, is a consequence of these two.

There is no official terms for the line at hand. Eric Weisstein refers to the line as the Nagel Line. I believe elsewhere I saw it being called Euler-Nagel line. Besides the 4 important points I, G, S, and M it houses a good deal of remarkable points, Kimberling's Encyclopedia, Line(1, 2, 8, 10, 42, ...).

The Nagel line is much less known than Euler's. While the latter appears practically in every book on Triangle Geometry, there are just so many sources that prove existence of the Nagel line. The most engaging is a recount by D. Hofstadter of his rediscovery of the line with the help of the dynamic geometry software.

### References

1. J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS - Chelsea Publishing, 1971, p. 55-57
2. D. R. Hofstadter, Discovery and Dissection of a Geometric Gem, in Geometry Turned On!, J. King and D. Schattschneider (editors), MAA, 1997, pp. 3-14
3. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995, pp. 5-13.
4. C. Kimberling, Triangle Centers and Central Triangles, Congr. Numer. 129, 1998, p. 128.