Gergonne Line


The Gergonne line is Oldknow's (1996) term for the perspectrix of a contact triangle DeltaDEF and its reference triangle DeltaABC. It is determined by the Nobbs points D^', E^', and F^'.


In addition to the Nobbs points, the Fletcher point (intersection with the Soddy line) and Evans point (intersection with the Euler line) also lie on the Gergonne line. Furthermore, the Soddy line and Gergonne line are perpendicular (Oldknow 1996).

The Gergonne line is central line L_(55) and has trilinear equation


where s is the semiperimeter of DeltaABC. It passes through Kimberling centers X_i for i=241, 514, 650, 665, 1323 (Fletcher point), and 1375 (Evans point), 1465, 1638, 3002, 3004, 3008, and 3015.

The Gergonne line is the perspectrix of the contact triangle and reference triangle, as well as the excentral triangle and medial triangle.


It is the radical line of the coaxal system (Bevan circle, excircles radical circle, polar circle) and (incircle, inner Soddy circle, outer Soddy circle).

The angle between the orthic axis and Gergonne line is equal to that between the Euler line and the Soddy line (F. Jackson, pers. comm., Nov. 2, 2005).

See also

Contact Triangle, Euler-Gergonne-Soddy Triangle, Euler Line, Evans Point, Fletcher Point, Nobbs Points, Soddy Line, Tangential Triangle

Explore with Wolfram|Alpha


Beauregard, R. A. and Suryanarayan, E. R. "Another Look at the Euler-Gergonne-Soddy Triangle." Math. Math. 76, 385-390, 2003.Oldknow, A. "The Euler-Gergonne-Soddy Triangle of a Triangle." Amer. Math. Monthly 103, 319-329, 1996.

Referenced on Wolfram|Alpha

Gergonne Line

Cite this as:

Weisstein, Eric W. "Gergonne Line." From MathWorld--A Wolfram Web Resource.

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