Three lines intersect forming a triangle of sides $(a, b, c) $.
Let $ab+bc+ca= \rho^2;$ Can we try to define a new third property along with $(r,R)$(in-circle, circumcircle radii) involving $\rho?$
Motivation is to find a new locus/curve with a geometric property alongwith $ (r,R) $ as another circle or curve. We naturally look to the coefficients in:
$$ (s-a)(s-b)(s-c)= s^3-s^2(a+b+c)+s(ab+bc+ca=\rho^2)-abc $$
Relation $ r^2+ 4rR + s^2 = =\rho^2$ is given in g.kov's comment.
Candidate characteristic "radius" length properties are:
$$[\dfrac{\rho^2}{r},\dfrac{\rho^2}{s},\dfrac{\rho^2}{R}]=(b_1,b_2,b_3)$$
$b_2$ curve is easy to calculate. It equals twice base of triangle $c$ between "foci" when sides $(a,b)$ describe a particular Oval of Cassini$ (e=2)$
$$\dfrac{ab+bc+ca}{a+b+c}=c\rightarrow ab=c^2;\;(ab+bc+ca)/s= 2 c ;\;$$
