Equal detour point

In Euclidean geometry, the equal detour point is a triangle center denoted by X(176) in Clark Kimberling's Encyclopedia of Triangle Centers. It is characterized by the equal detour property: if one travels from any vertex of a triangle ABC to another by taking a detour through some inner point P, then the additional distance traveled is constant. This means the following equation has to hold:[1]

  Triangle ABC (side lengths a, b, c)
  Incircle (centered at incenter I)
  Isoperimetric lines dA, dB, dC (concur at isoperimetric point Q)
  Detours hA, hB, hC (concur at equal detour point P):
I, Q, P and the Gergonne point G are collinear and form a harmonic range:

The equal detour point is the only point with the equal detour property if and only if the following inequality holds for the angles α, β, γ of ABC:[2]

If the inequality does not hold, then the isoperimetric point possesses the equal detour property as well.

The equal detour point, isoperimetric point, the incenter and the Gergonne point of a triangle are collinear, that is all four points lie on a common line. Furthermore, they form a harmonic range as well (see graphic on the right).[3]

The equal detour point is the center of the inner Soddy circle of a triangle and the additional distance travelled by the detour is equal to the diameter of the inner Soddy Circle.[3]

The barycentric coordinates of the equal detour point are[3]

and the trilinear coordinates are:[1]

References

  1. Isoperimetric point and equal detour point at the Encyclopedia of Triangle Centers (retrieved 2020-02-07)
  2. M. Hajja, P. Yff: "The isoperimetric point and the point(s) of equal detour in a triangle". Journal of Geometry,November 2007, Volume 87, Issue 1–2, pp 76–82, https://doi.org/10.1007/s00022-007-1906-y
  3. N. Dergiades: "The Soddy circles" Forum Geometricorum volume 7, pp. 191–197, 2007
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