A Peculiar Locus Area

Calculus Level 3

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

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

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

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Daniel Liu by Daniel Liu
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