Point of the same detour

isoperimetric point Q, inscribed center point I, Gergonne point G, point same detour P
same detours: Harmonic relationship:
${\displaystyle h_{A}+h_{C}-b=h_{A}+h_{B}-c=h_{B}+h_{C}-a}$ ${\displaystyle {\frac {|QI|}{|PI|}}={\frac {|QG|}{|PG|}}}$

The point of the same detour is a special point in a triangle . This point is characterized by the fact that the detour from via to is just as great as the detour from via to and the detour from via to . Here, the length of the detour is understood to be the length of the route that is covered in addition to the shortest connection and accordingly: ${\displaystyle ABC}$ ${\displaystyle P}$ ${\displaystyle A}$ ${\displaystyle P}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle P}$ ${\displaystyle C}$ ${\displaystyle B}$ ${\displaystyle P}$ ${\displaystyle C}$

${\displaystyle {\begin{aligned}&|AP|+|PC|-|AC|\\=&|AP|+|PB|-|AB|\\=&|BP|+|PC|-|BC|\end{aligned}}}$.

Uniqueness

The point of the same detour has the Kimberling number X (176). It is the only point with the above property if the following inequality is true for the angles of the triangle : ${\displaystyle \alpha ,\beta ,\gamma }$ ${\displaystyle ABC}$

${\displaystyle \tan \left({\frac {\alpha }{2}}\right)+\tan \left({\frac {\beta }{2}}\right)+\tan \left({\frac {\gamma }{2}}\right)\leq 2}$.

If the inequality is not fulfilled, the isoperimetric point also has the property of the same detour.

properties

• The point of the same detour is harmonically related to the isoperimetric point, the center of the inscribed circle and the Gergonne point and collinear to these three points.
• The detours are equal to the diameter of the inner Soddy circle .
• The barycentric coordinates are:

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

Here stands for the area and for half the circumference of the triangle . ${\displaystyle \Delta }$ ${\displaystyle s}$ ${\displaystyle ABC}$

• The trilinear coordinates are:

${\displaystyle \left(\sec \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)+1:\sec \left({\frac {\beta }{2}}\right)\cos \left({\frac {\gamma }{2}}\right)\cos \left({\frac {\alpha }{2}}\right)+1:\sec \left({\frac {\gamma }{2}}\right)\cos \left({\frac {\alpha }{2}}\right)\cos \left({\frac {\beta }{2}}\right)+1\right)}$.

Web links

Commons : point of the same detour  - collection of images, videos and audio files

• isoperimetric and equal detour points - interactive illustration on Geogebratube

Individual evidence

1. ↑ ^{a b} Isoperimetric point and equal detour point in the Encyclopedia of Triangle Centers (accessed February 7)
2. ↑ M. Hajja, P. Yff: The isoperimetric point and the point (s) of equal detour in a triangle . In: Journal of Geometry , November 2007, Volume 87, Issue 1–2, pp. 76–82, https://doi.org/10.1007/s00022-007-1906-y
3. ↑ ^{a b c} N. Dergiades: The Soddy circles . In: Forum Geometricorum , Volume 7, pp. 191–197, 2007