Question:

Step 1: Identify the geometric configuration and the relevant formula for the distance. Three circles touch one another externally. Let their radii be $r_1, r_2, r_3$. The centers of these circles, $O_1, O_2, O_3$, form a triangle. The side lengths of this triangle are the sums of the radii: $a = r_2+r_3$, $b = r_1+r_3$, and $c = r_1+r_2$. The tangents at their points of contact meet at a single point, which is the radical center of the three circles. It is a known geometric property that the distance from this radical center to any of the points of contact is equal to the inradius of the triangle formed by the centers of the three circles. Let this distance be $d$. So, $d = r_{in}$, where $r_{in}$ is the inradius of $\triangle O_1O_2O_3$. The problem states that $d=4$. Step 2: Calculate the semi-perimeter and area of the triangle formed by the centers. Let $s$ be the semi-perimeter of $\triangle O_1O_2O_3$. $$s = \frac{(r_1+r_2) + (r_2+r_3) + (r_3+r_1)}{2} = \frac{2(r_1+r_2+r_3)}{2} = r_1+r_2+r_3$$ Let $K$ be the area of $\triangle O_1O_2O_3$. Using Heron's formula $K = \sqrt{s(s-a)(s-b)(s-c)}$, we substitute the values: $$s-a = (r_1+r_2+r_3) - (r_2+r_3) = r_1$$ $$s-b = (r_1+r_2+r_3) - (r_1+r_3) = r_2$$ $$s-c = (r_1+r_2+r_3) - (r_1+r_2) = r_3$$ Therefore, the area $K$ is: $$K = \sqrt{(r_1+r_2+r_3)r_1r_2r_3}$$ Step 3: Determine the inradius of the triangle formed by the centers. The inradius $r_{in}$ of a triangle is given by the formula $r_{in} = \frac{\text{Area}}{\text{Semi-perimeter}}$. Substituting the expressions for $K$ and $s$: $$r_{in} = \frac{\sqrt{r_1r_2r_3(r_1+r_2+r_3)}}{r_1+r_2+r_3}$$ This expression can be simplified by recognizing that $r_1+r_2+r_3 = \sqrt{(r_1+r_2+r_3)^2}$: $$r_{in} = \sqrt{\frac{r_1r_2r_3(r_1+r_2+r_3)}{(r_1+r_2+r_3)^2}} = \sqrt{\frac{r_1r_2r_3}{r_1+r_2+r_3}}$$ This matches the formula for $r$ when "properly simplified for the triangle configuration shown" mentioned in the original solution's context (if $a,b,c$ were $r_1,r_2,r_3$ and the initial formula was correct). Step 4: Use the given distance to establish a relationship between the radii. We are given that the distance from the point of contact is $4$. Therefore, $r_{in} = 4$. $$4 = \sqrt{\frac{r_1r_2r_3}{r_1+r_2+r_3}}$$ Squaring both sides of the equation to remove the square root: $$4^2 = \frac{r_1r_2r_3}{r_1+r_2+r_3}$$ $$16 = \frac{r_1r_2r_3}{r_1+r_2+r_3}$$ Step 5: Calculate the required ratio. The problem asks for the ratio of the product of the radii to the double of the sum of the radii. This can be written as: $$\text{Ratio} = \frac{r_1r_2r_3}{2(r_1+r_2+r_3)}$$ We can rewrite this expression by factoring out $\frac{1}{2}$: $$\text{Ratio} = \frac{1}{2} \times \left( \frac{r_1r_2r_3}{r_1+r_2+r_3} \right)$$ From Step 4, we know that $\frac{r_1r_2r_3}{r_1+r_2+r_3} = 16$. Substitute this value into the ratio expression: $$\text{Ratio} = \frac{1}{2} \times 16$$ $$\text{Ratio} = 8$$ The final answer is $\boxed{8}$.