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)

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.

CITE THIS AS:

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