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] 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 (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]

γ

Triangle △ ABC (side lengths a, b, c )
Incircle (centered at incenter I )
Isoperimetric lines d A , d B , d C (concur at isoperimetric point Q )
Detours h A , h B , h C (concur at equal detour point P ):
I, Q, P and the Gergonne point G are collinear and form a harmonic range :