MathNet 1994 — Part 01 Question 51

Let $O$ be an interior point in the equilateral triangle $ABC$, of side length $a$. The lines $AO$, $BO$, and $CO$ intersect the sides of the triangle in the points $A_1$, $B_1$, and $C_1$. Show that
$$
|OA_1| + |OB_1| + |OC_1| \lt a
$$