Gergonne Line

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

$$D = B+{f\over e}C$$

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

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

References

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