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Problem of the Day #20: Kissing Circles April 8, 2011

Posted by Saketh in : potd , trackback

Radii $r_1$, $r_2$, and $r_3$ are chosen independently and uniformly at random from the interval $[0,1]$. Three mutually tangent circles $A$, $B$, and $C$ are then drawn using these radii. A fourth circle $D$ is drawn so that each of $A$, $B$, and $C$ is internally tangent to $D$.

For any pair of circles chosen from the set $\{A,B,C\}$, a new circle can be drawn between the selected pair and circle $D$ so that they are all mutually tangent. Let the radii of the new circles be $r_{ab}$, $r_{bc}$, and $r_{ac}$.

Determine the expected value of the product $r_{ab} \cdot r_{bc} \cdot r_{ac}$.

Bonus: Suppose we know that $r_1$ is at least twice $r_2$. Determine the expected value of $r_{ab} \cdot r_{bc} \cdot r_{ac}$.


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