MathNet 2024-part-04 — 168 problems (EN–VI) Question 150

Let $ABCDA'B'C'D'$ be a cube and $O$ its centre. Consider the points $M \in [AD']$, $N \in [B'C]$ and $P$ a point on the face $A'B'C'D'$. Denote $E$ and $F$ the midpoints of the edges $A'B'$, $C'D'$. Prove that $O$ is the barycentre of the triangle $PMN$ if and only if $P \in [EF]$ and $\frac{PF}{AM} = \frac{PE}{CN} = \sqrt{2}$.
Traian Preda