Optimization & Lagrange Multipliers · Superchallenge problem

A right triangle has legs $3$ and $4$ and hypotenuse $5$. For a point $P$ inside it, let $d_1$, $d_2$, $d_3$ be the distances from $P$ to the three sides. Find the minimum possible value of $d_1^2 + d_2^2 + d_3^2$.