VN-TST 2003 Problem 4

On the sides of triangle $ABC$ take the points $M_1, N_1, P_1$ such that each line $MM_1, NN_1, PP_1$ divides the perimeter of $ABC$ in two equal parts ( $M, N, P$ are respectively the midpoints of the sides $BC, CA, AB$ ).

I. Prove that the lines $MM_1, NN_1, PP_1$ are concurrent at a point $K$ . II. Prove that among the ratios $\frac{KA}{BC}, \frac{KB}{CA}, \frac{KC}{AB}$ there exist at least a ratio which is not less than $\frac{1}{\sqrt{3}}$ .