Tangent of Locus

Geometry Level pending

\(ABC\) is a triangle with \(AB\) on the horizontal axis, \(\angle ABC=75^\circ\) and \(\angle BAC=15^\circ.\)

Now, we move around vertex \(C\) such that the sum of triangle \(ABC\)'s perimeter and its altitude dropped from \(C\) to \(BA\) remains constant, and trace the locus of \(C.\)

Let \(CD\) be the tangent line to the locus at \(C\) above. If the slope of this tangent line, \(m,\) is such that \[ \dfrac 1m = \sqrt P + \sqrt Q ,\] where \(P \) and \(Q\) are positive integers, what is \(P + Q?\)

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