RO-NMO 2021 Problem 81

In the cuboid $ABCDA'B'C'D'$ with $AB=a$ , $AD=b$ and $AA'=c$ such that $a>b>c>0$ , the points $E$ and $F$ are the orthogonal projections of $A$ on the lines $A'D$ and $A'B$ , respectively, and the points $M$ and $N$ are the orthogonal projections of $C$ on the lines $C'D$ and $C'B$ , respectively. Let $DF\cap BE=\{G\}$ and $DN\cap BM=\{P\}$ . [list=a] [] Show that $(A'AG)\parallel (C'CP)$ and determine the distance between these two planes; [] Show that $GP\parallel (ABC)$ and determine the distance between the line $GP$ and the plane $(ABC)$ . [/list]

Petre Simion, Nicolae Victor Ioan