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 to another by taking a detour through some inner point, then the additional distance traveled is constant. This means the following equation has to hold:[1]
\begin{align} &\overline{AP}+\overline{PC}-\overline{AC}\\[3mu] ={}&\overline{AP}+\overline{PB}-\overline{AB}\\[3mu] ={}&\overline{BP}+\overline{PC}-\overline{BC}. \end{align}
The equal detour point is the only point with the equal detour property if and only if the following inequality holds for the angles of :[2]
\tan\tfrac12\alpha+\tan\tfrac12\beta+\tan\tfrac12\gamma\leq2
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]
\left(a+
| \Delta | |
| s-a |
:b+
| \Delta | |
| s-b |
:c+
| \Delta | |
| s-c |
\right).