Isoperimetric point

isoperimetric point P, inscribed center point I, Gergonne point G, point of the same detour T

The isoperimetric point is a distinguished point in a triangle ABC. It is the point P in this triangle, for which the partial triangles PBC, PCA and PAB have the same circumference . The isoperimetric point has the kimberling number X (175).

properties

• The isoperimetric point is harmonically related to the point of the same detour with respect to the center of the circle and the Gergonne point and thus collinear to these three points.
• The circumferences of PBC, PCA and PAB are equal to the diameter of the outer Soddy circle . ${\displaystyle {\tfrac {2\triangle }{\left\vert 4R+r-(a+b+c)\right\vert }}}$
• The isoperimetric point exists if and only if the circumference of ABC is greater than , where is the radius of the perimeter and the radius of the inscribed circle . ${\displaystyle 4R+r}$ ${\displaystyle R}$ ${\displaystyle r}$

Coordinates

Isoperimetric point ( ) ${\displaystyle X_{175}}$
Trilinear coordinates	${\displaystyle \left(\sec {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}-1\right):\left(\sec {\frac {\beta }{2}}\cos {\frac {\gamma }{2}}\cos {\frac {\alpha }{2}}-1\right):\left(\sec {\frac {\gamma }{2}}\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}-1\right)}$
Barycentric coordinates	${\displaystyle \left(a-{\frac {\Delta }{s-a}}\right):\left(b-{\frac {\Delta }{s-b}}\right):\left(c-{\frac {\Delta }{s-c}}\right)}$

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

literature

• GR Veldkamp: The Isoperimetric Point and the Point (S) of Equal Detour in a Triangle . In: The American Mathematical Monthly , Volume 92, No. 8, Oct. 1985, pp. 546–558 ( JSTOR 2323159 )

Web links

• Isoperimetric punt (Dutch)
• C. Kimberling: Isoperimetric Point And Equal Detour Point
• Eric W. Weisstein : Isoperimetric Point . In: MathWorld (English).
• isoperimetric and equal detour points - interactive illustration on GeoGebratube