Given Triangle, Any Side, Any Point!

Geometry Level 5

The lengths of the sides of a triangle $$ABC$$ are $$4,5$$ and $$6$$. Take any point $$D$$ on any one of the sides of the triangle and drop perpendiculars $$DP$$ and $$DQ$$ onto the two other sides ($$P$$ and $$Q$$ are on the sides). Let the minimum possible value for the length of $$PQ$$ be $$L$$.

If $$L$$ can be expressed as $$\dfrac{A}{B}$$ for positive coprime integers $$A,B$$, then submit the value of $$A+B$$ as your answer.

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