Isoperimetric Point

The point $S'$ which makes the Perimeters of the Triangles $\Delta BS'C$, $\Delta CS'A$, and $\Delta AS'B$ equal. The isoperimetric point exists Iff the largest Angle of the triangle satisfies

$$\max(A,B,C) < 2\sin^{-1}({\textstyle{4\over 5}})\approx 1.85459{\rm\ rad}\approx 106.26^\circ,$$

or equivalently

$$a+b+c>4R+r,$$

where $a$, $b$, and $c$ are the side lengths of $\Delta ABC$, $r$ is the Inradius, and $R$ is the Circumradius. The isoperimetric point is also the center of the outer Soddy Circle of $\Delta ABC$ and has Triangle Center Function

$$\alpha=1-{2\Delta\over a(b+c-a)} = \sec({\textstyle{1\over 2}}A)\cos({\textstyle{1\over 2}}B)\cos({\textstyle{1\over 2}}C)-1.$$

See also Equal Detour Point, Perimeter, Soddy Circles

References

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' Math. Mag. 67, 163-187, 1994.

Kimberling, C. ``Isoperimetric Point and Equal Detour Point.'' http://cedar.evansville.edu/~ck6/tcenters/recent/isoper.html.

Kimberling, C. and Wagner, R. W. ``Problem E 3020 and Solution.'' Amer. Math. Monthly 93, 650-652, 1986.

Veldkamp, G. R. ``The Isoperimetric Point and the Point(s) of Equal Detour.'' Amer. Math. Monthly 92, 546-558, 1985.