Equal detour point

Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line $$\begin{array}{cc}& h_A+h_C-b\\ =& h_A+h_B-c\\ =& h_B+h_C-a\end{array}$$ Template:Mvar and the Gergonne point Template:Mvar are collinear and form a harmonic range: $$\frac{\overline{QI}}{\overline{PI}}=\frac{\overline{QG}}{\overline{PG}}$$

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 Template:Math to another by taking a detour through some inner point Template:Mvar, then the additional distance traveled is constant. This means the following equation has to hold:

$\begin{array}{cc}& \overline{AP}+\overline{PC}-\overline{AC}\\ =& \overline{AP}+\overline{PB}-\overline{AB}\\ =& \overline{BP}+\overline{PC}-\overline{BC}.\end{array}$

The equal detour point is the only point with the equal detour property if and only if the following inequality holds for the angles Template:Mvar of Template:Math:

$\tan \frac{1}{2}\alpha +\tan \frac{1}{2}\beta +\tan \frac{1}{2}\gamma \le 2$

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).

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.

The barycentric coordinates of the equal detour point are

$(a+\frac{\mathrm{\Delta }}{s-a}:b+\frac{\mathrm{\Delta }}{s-b}:c+\frac{\mathrm{\Delta }}{s-c}).$

and the trilinear coordinates are: $$1+\frac{\cos \frac{1}{2}\beta \cos \frac{1}{2}\gamma }{\cos \frac{1}{2}\alpha }:1+\frac{\cos \frac{1}{2}\gamma \cos \frac{1}{2}\alpha }{\cos \frac{1}{2}\beta }:1+\frac{\cos \frac{1}{2}\alpha \cos \frac{1}{2}\beta }{\cos \frac{1}{2}\gamma }$$

External links

• isoperimetric and equal detour points – interaktive Illustration auf Geogebratube