P and the Triangle

Geometry Level 5

Let \(P\) be a point in triangle \(ABC\) with area \(1\), and let \(f(P)=PA+PB+PC\). For every triangle \(ABC\), let \(h_{ABC}=\min_P\big[f(P)\big]\).

Find \(\displaystyle \min_{ABC}\left[h_{ABC}\right]^4.\)


Inspiration.

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