# 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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