# Three Lines, Three Circles

Geometry Level 5

Lines $$l, m, n$$ in the same plane are such that $$l \cap m = A,$$ $$m \cap n = B,$$ $$l \cap n = C,$$ and $$A \neq B \neq C.$$ The three circles that are tangent to $$l, m,$$ and $$n$$ but not in the interior of $$ABC$$ have radii $$3, 4,$$ and $$5.$$ If $$AB + BC + CA = p\sqrt{q},$$ for positive integers $$p$$ and $$q$$ such that $$q$$ is not divisible by the square of any prime, find $$p+q.$$

Notes:

Assume that such a configuration described in the problem is possible.

$$a \cap b$$ means the intersection of lines $$a$$ and $$b.$$

See the next problem here.

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