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Tangent Circles

Source: AIME 2008II Problem 11

April 3, 2008
trigonometryquadraticsgeometryprobabilityAMCAIMEtrapezoid

Problem Statement

In triangle ABC ABC, AB \equal{} AC \equal{} 100, and BC \equal{} 56. Circle P P has radius 16 16 and is tangent to AC‾ \overline{AC} and BC‾ \overline{BC}. Circle Q Q is externally tangent to P P and is tangent to AB‾ \overline{AB} and BC‾ \overline{BC}. No point of circle Q Q lies outside of △ABC \triangle ABC. The radius of circle Q Q can be expressed in the form m \minus{} n\sqrt {k}, where m m, n n, and k k are positive integers and k k is the product of distinct primes. Find m \plus{} nk.
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