Suppose a triangle *T* has sides *a*, *b*, and *c*.

Let *s* be the semi-perimeter, i.e. half the perimeter.

Let *r* be the inner radius, the radius of the largest circle that can fit inside *T*.

Let *R* be the outer radius, the radius of the smallest circle that can enclose *T*.

Then three simple equations relate *a*, *b*, *c*, *s*, *r*, and *R*.

Given *a*, *b*, and *c*, use the first equation to solve for *s*, then the third equation for *Rr*, then the second for *r*, then go back to the last equation to find *R*.

Given *s*, *r*, and *R*, you can calculate the right hand sides of the three equations above, which are the coefficients in a cubic equation for the sides *a*, *b*, and *c*.

Note that this last statement is not about triangles per se. It’s a consequence of

which would be true even if *a*, *b*, and *c* were not the sides of a triangle. But since they *are* sides of a triangle here, the coefficients can be interpreted in terms of geometry, namely in terms of perimeter, inner radius, and outer radius.