Three Lines, Three Circles

Geometry Level 5

Lines l,m,nl, m, n in the same plane are such that lm=A,l \cap m = A, mn=B,m \cap n = B, ln=C,l \cap n = C, and ABC.A \neq B \neq C. The three circles that are tangent to l,m,l, m, and nn but not in the interior of ABCABC have radii 3,4,3, 4, and 5.5. If AB+BC+CA=pq,AB + BC + CA = p\sqrt{q}, for positive integers pp and qq such that qq is not divisible by the square of any prime, find p+q.p+q.

Notes:

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

aba \cap b means the intersection of lines aa and b.b.

See the next problem here.

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