Equal detour point
![](http://upload.wikimedia.org/wikipedia/commons/thumb/0/01/Equal_detour_point.svg/330px-Equal_detour_point.svg.png)
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]
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
[edit]- ^ a b Isoperimetric point and equal detour point at the Encyclopedia of Triangle Centers (retrieved 2020-02-07)
- ^ 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
- ^ a b c N. Dergiades: "The Soddy circles" Forum Geometricorum volume 7, pp. 191–197, 2007
External links
[edit]![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
- isoperimetric and equal detour points – interaktive Illustration auf Geogebratube