Challenge Your Brain 2

Geometry Level 5

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

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