74th Romanian Mathematical Olympiad

Let $ABC{A}^{\prime }{B}^{\prime }{C}^{\prime }$ be a triangular regular prism, with lateral edges $A{A}^{\prime }$, $B{B}^{\prime }$, $C{C}^{\prime }$. Consider the midpoint $D$ of the edge $BC$ and the parallelogram $AD{B}^{\prime }E$. Let $F$ be the orthogonal projection of the point $A^{\prime }$ on the line $AE$, $d$ be the intersection of the planes $(ADE)$ and $(A^{\prime }CF)$ and $P$ be the intersection of the line $d$ with the plane $(ABC)$. Prove that $P$ is the baricentre of the triangle $ABC$ if and only if $AB=A{A}^{\prime }\sqrt{2}$. Valeriu Bărbieru