MathNet 2025 — batch of 15 questions Question 14

Let $\triangle ABC$ be an equilateral triangle with side length $4$. Across all points $P$ inside triangle $\triangle ABC$ satisfying $[PAB] + [PAC] = [PBC]$, compute the minimum possible length of $PA$.

(Here, $[XYZ]$ denotes the area of triangle $\triangle XYZ$.)