AUSTRIAN-POLISH 1982 Problem 8

Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$ . Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$ . Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$ .