Let $A, B, C$ be centers of three circles that are mutually tangent externally, let $r_A, r_B, r_C$ be the radii of the circles, respectively. Let $r$ be the radius of the incircle of $\vartriangle ABC$. Prove that $$r^2 \le \frac19 (r_A^2 + r_B^2+r_C^2)$$and identify, with justification, one case where the equality is attained.