Question 130
Let $\triangle ABC$ be an equilateral triangle and let $A_1, A_2$ be points on the side $BC$, $B_1, B_2$ be points on the side $CA$, $C_1, C_2$ be points on the side $AB$, such that $BA_1 \lt BA_2$, $CB_1 \lt CB_2$, $AC_1 \lt AC_2$ and $A_1A_2 = B_1B_2 = C_1C_2$. Prove that there exists a triangle whose side lengths are $A_2B_1$, $B_2C_1$, $C_2A_1$.