VN TST 2003 Question 4

Let $M,N,P$ be the midpoints of $BC,CA,AB$ respectively in triangle $ABC$. Let $M_1,N_1,P_1$ be points on its perimeter such that each of $MM_1,NN_1,PP_1$ bisects the perimeter. (a) Prove that these lines have a common point $K$. (b) Show that at least one of $KA/BC$, $KB/CA$, $KC/AB$ is not less than $1/\sqrt3$.

Here “bisects the perimeter” means that the two boundary arcs between a pair of endpoints have equal length.