Take a large enough triangle. Let $R = \max \{r_1+r_2, r_2+r_3, r_3+r_1 \}$. Construct a equilateral triangle with length $R$ and vertices $p_1 = (0,0), p_2 = (R,0), p_3 = (\frac{1}{2}R,\frac{\sqrt{3}}{2}R )$.

Then take the usual self-similar maps $f_i(x)=r_i(x−p_i)+p_i, 1 \leq i \leq 3$, and take $O$ to be the interior of the triangle.

Corrected:

Let $R = \max \{r_1+r_2, r_2+r_3, r_3+r_1 \}$. This can be done when $R \leq 1$.

Take 3 non-collinear points $p_1,p_2,p_3$ on the plane and take the usual self-similar maps $f_i(x)=r_i(x−p_i)+p_i, 1 \leq i \leq 3$, and let $O$ be the interior of the triangle formed by the 3 points.

Take a large enough triangle. Let $R = \max(r_1+r_2, r_2+r_3, r_3+r_1)$. Construct a equilateral triangle with length $R$ with vertices $p_1 = (0,0), p_2 = (R,0), p_3 = (\frac{1}{2}R,\frac{\sqrt{3}}{2}R )$.

Then take the usual self-similar maps $f_i(x)=r_i(x−p_i)+p_i, 1 \leq i \leq 3$, and take $O$ to be the interior of the triangle.

Take a large enough triangle. Let $R = \max \{r_1+r_2, r_2+r_3, r_3+r_1 \}$. Construct a equilateral triangle with length $R$ and vertices $p_1 = (0,0), p_2 = (R,0), p_3 = (\frac{1}{2}R,\frac{\sqrt{3}}{2}R )$.

Then take the usual self-similar maps $f_i(x)=r_i(x−p_i)+p_i, 1 \leq i \leq 3$, and take $O$ to be the interior of the triangle.