NMC 1994 Problem 1

Let $O$ be an interior point in the equilateral triangle $A B C$, of side length $a$. The lines $A O, B O$, and $C O$ intersect the sides of the triangle in the points $A_{1}, B_{1}$, and $C_{1}$. Show that

$$\left|O A_{1}\right|+\left|O B_{1}\right|+\left|O C_{1}\right|<a$$