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Nobbs' Points, Gergonne Line: What are they?
A Mathematical Droodle


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Explanation

Copyright © 1996-2008 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Nobbs' Points, Gergonne Line

The applet attempts to illustrate the following theorem:

Assume A', B', C' are the points of contact of the incircle of ABC: A' on side BC, etc. Denote the intersection of AB and A'B' as C'', that of AC and A'C' as B'', and let A'' be the intersection of BC and B'C'. Then points A'', B'', and C'' are collinear.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


Buy this applet

The points A'', B'', and C'' are known as Nobbs' points. The theorem tells us the Nobbs' points of a triangle are collinear. The line the points lie on was named the Gergonne line by Adrian Oldknow [Oldknow, pp. 324-325] after J. D. Gergonne. The reason for the name is as follows.

It is known that the lines AA', BB', and CC' meet at the Gergonne point Ge of ABC. Put another way, this means that two triangles ABC and A'B'C' are perspective in Ge. By Desargues' theorem, triangles perspective from a point are also perspective from a line. Thus the corresponding side lines of triangles ABC and A'B'C' cross in the points that belong to the same line. This is the Gergonne line of Oldknow.

I'll give an additional argument in support of that nomenclature: the Gergonne line is the polar of the Gergonne point Ge in the incircle of ABC. Indeed,

 
A'B'is the polar ofC, 
ABis the polar ofC'. Therefore,
C''is the pole ofCC', by La Hire's theorem.

Similarly, A'' is the pole of AA' and B'' is the pole of BB'. Since their polars AA', BB', and CC' are concurrent (in the Gergonne point Ge), the poles A'', B'', and C'' are collinear to a line (the Gergonne line, naturally.)

Note that if one of the triangles ABC and A'B'C' is equilateral, so is the other, and in this case the Gergonne line is the line at infinity.

Remark

A'B'C' is known as the contact triangle (and also Gergonne triangle) of ABC. ABC is the tangent(ial) triangle of A'B'C'.

(From a somewhat different perspective this same configuration is studied elsewhere. A more general statement is also available.)

References

  1. A. Oldknow, The Euler-Gergonne-Soddy Triangle of a Triangle, Amer Math Monthly, Vol. 103, No. 4 (Apr. 1996), 319-329

Poles and Polars

Desargues' Theorem

Copyright © 1996-2008 Alexander Bogomolny

28676795Page copy protected against web site content infringement by Copyscape


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