KöMaL 2013 B4548

ABC is an isosceles right angled triangle of unit legs. $A_1$ is a point on side $AB$, $B_1$ is a point on side $BC$, and $C_1$ is a point on hypotenuse $CA$. What is the minimum possible length of $A_1 B_1$ if the triangles ABC and $A_1 B_1 C_1$ are similar?