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r^2 <= 1/9 (r_A^2 + r_B^2+r_C^2), 3 mutually tangent externally circles

Source: Thailand Mathematical Olympiad 2015 p7

August 16, 2020
geometric inequalitygeometrycirclestangent circlesradiiinradius

Problem Statement

Let A,B,CA, B, C be centers of three circles that are mutually tangent externally, let rA,rB,rCr_A, r_B, r_C be the radii of the circles, respectively. Let rr be the radius of the incircle of △ABC\vartriangle ABC. Prove that r2≤19(rA2+rB2+rC2)r^2 \le \frac19 (r_A^2 + r_B^2+r_C^2) and identify, with justification, one case where the equality is attained.
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