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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


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


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.''

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.

© 1996-9 Eric W. Weisstein