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Problem of the Day #100: Tangent Circles June 27, 2011

Posted by Billy in : potd , trackback

Circle $A$ has radius $3$ and circle $B$ has radius $2$. Their centers are $13$ units from one another. A common internal tangent $\overline{PQ}$ is drawn such that $P$ lies on $A$ and $Q$ lies on $B$. Next, circles $A’$ and $B’$ are constructed outside of circles $A$ and $B$ such that circle $A’$ is tangent to $\overline{AB}$, $\overline{PQ}$, and circle $A$, and circle $B’$ is tangent to $\overline{AB}$, $\overline{PQ}$, and circle $B$. The distance between the centers of $A’$ and $B’$ can be expressed in the form $\frac{a\sqrt{b}-c\sqrt{d}}{e}$, where $a$, $c$, and $e$ are relatively prime and $b$ and $d$ are not divisible by the square of any prime. Find $a+b+c+d+e$.


1. Eugene - June 27, 2011

Is A’B’ the shortest distance between the circles or the distance between their centers?

2. Billy - June 27, 2011

The centers. I’ll update the post to make that clearer.