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Isoperimetric Point


IsoperimetricPoint

The point S^' which makes the perimeters of the triangles DeltaBS^'C, DeltaCS^'A, and DeltaAS^'B equal. The isoperimetric point exists iff

 a+b+c>4R+r,
(1)

where a, b, and c are the side lengths of DeltaABC, r is the inradius, and R is the circumradius. The isoperimetric point is also the center of the outer Soddy circle of DeltaABC and has equivalent triangle center functions

alpha=1-(2Delta)/(a(b+c-a))
(2)
alpha=sec(1/2A)cos(1/2B)cos(1/2C)-1.
(3)

See also

Equal Detour Point, Perimeter, Soddy Circles

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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://faculty.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.

Referenced on Wolfram|Alpha

Isoperimetric Point

Cite this as:

Weisstein, Eric W. "Isoperimetric Point." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IsoperimetricPoint.html

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