A Peculiar Locus Area

Calculus Level 3

ABC\triangle ABC is constructed such that AB=3AB=3, BC=4BC=4, and ABC=90\angle ABC=90^{\circ}. Point PP is chosen inside ABC\triangle ABC, and points EE and FF are drawn such that they form line segments with PP that are perpendicular to sides ABAB and BCBC, respectively. If L\mathbf{L} is the locus of all points PP such that [PEBF]1[PEBF]\ge 1, then find the value of 1000[L][ABC]\left\lfloor 1000\dfrac{[\mathbf{L}]}{[ABC]}\right\rfloor

Details and Assumptions:\text{Details and Assumptions:}

[N][\mathbf{N}] means the area of the locus N\mathbf{N}, and [PQRS][PQRS] means the area of PQRSPQRS. x\lfloor x \rfloor is the floor function.

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