TOPICS

# Isoperimetric Point

The point which makes the perimeters of the triangles , , and equal. The isoperimetric point exists iff

 (1)

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

 (2) (3)

## See also

Equal Detour Point, Perimeter, Soddy Circles

## Explore with Wolfram|Alpha

More things to try:

## 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