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Gergonne Line

\begin{figure}\begin{center}\BoxedEPSF{NobbsPoints.epsf scaled 900}\end{center}\end{figure}

The perspective line for the Contact Triangle $\Delta DEF$ and its Tangential Triangle $\Delta ABC$. It is determined by the Nobbs Points $D'$, $E'$, and $F'$. In addition to the Nobbs Points, the Fletcher Point and Evans Point also lie on the Gergonne line where it intersects the Soddy Line and Euler Line, respectively. The $D$ and $D'$ coordinates are given by

$\displaystyle D$ $\textstyle =$ $\displaystyle B+{f\over e}C$  
$\displaystyle D'$ $\textstyle =$ $\displaystyle B-{f\over e}C,$  

so $BDCD'$ form a Harmonic Range. The equation of the Gergonne line is

{\alpha\over d}+{\beta\over e}+{\gamma\over f}=0.

See also Contact Triangle, Euler Line, Evans Point, Fletcher Point, Nobbs Points, Soddy Line, Tangential Triangle


Oldknow, A. ``The Euler-Gergonne-Soddy Triangle of a Triangle.'' Amer. Math. Monthly 103, 319-329, 1996.

© 1996-9 Eric W. Weisstein