[ Extreme properties of a triangle (miscellaneous).]

The area of triangle $ABC$ is 1. Let $A_1, B_1, C_1$ be the midpoints of sides $BC, CA, AB$ respectively. Points $K, L, M$ are taken on segments $AB_1, CA_1, BC_1$ respectively. What is the minimum area of the common part of triangles $KLM$ and $A_1B_1C_1$?