Let $x,y,z$ the distances of the orhtocenter $H$ from $BC,CA,AB$ resp . we know the power of $H$ wrt the circumcircle is $2x.HA=2y.HB=2z.HC\implies 4Rx\cos A=4Ry\cos B=4Rz\cos C$ but since $ABC$ is acute the order of $\cos A ,\cos B,\cos C$ is inverse wrt $\angle A,\angle B,\angle C$ so $x,y,z$ and $a,b,c$ have the same order thus $ax+by+cz\ge \frac1 3 (a+b+c)(x+y+z) \implies r(a+b+c)\ge \frac1 3 (a+b+c)(x+y+z) \implies x+y+z \le 3 r$