Final Round of National Olympiad

The midpoints of sides $C_2{C}_3$, $C_3{C}_1$ and $C_1{C}_2$ of a triangle $C_1{C}_2{C}_3$ are $K_1$, $K_2$ and $K_3$, respectively. The centers of circles $c_1$, $c_2$ and $c_3$ are $C_1$, $C_2$ and $C_3$, respectively, and the centers of circles $k_1$, $k_2$, $k_3$ are $K_1$, $K_2$, $K_3$, respectively. No two of the given six circles intersect in two points nor are they inside each other. Circles $k_1$, $k_2$ and $k_3$ touch each other externally.

i) Prove that the sum of the radii of circles $c_1$, $c_2$ and $c_3$ does not exceed one quarter of the perimeter of the triangle $C_1{C}_2{C}_3$.

ii) Prove that if the sum of the radii of circles $c_1$, $c_2$ and $c_3$ equals one quarter of the perimeter of the triangle $C_1{C}_2{C}_3$ then the triangle $C_1{C}_2{C}_3$ is equilateral.