Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from point of contact is $4$. Find the ratio of the product of the radii to the double of the sum of the radii of the circles.
Step-by-Step Solution
Key Concept: The inradius formula $r = \frac{A}{s}$ combined with Heron's formula relates the inradius directly to the sides and semiperimeter.
For a triangle with sides $a, b, c$ and area $S$, the inradius is $r = \frac{A}{S}$. Using Heron's formula $S = \sqrt{abc(a+b+c)}$, we get $r = \frac{abc}{4\sqrt{abc(a+b+c)}} = \frac{\sqrt{abc}}{2\sqrt{a+b+c}}$. This can be rewritten as $r = \frac{abc}{2(a+b+c)}$ when properly simplified for the triangle configuration shown.
Correct Answer: 8