Problem

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Tags: geometry, 3D geometry, parallelepiped



$ABCDA'B'CD'$ is a rectangular parallelepiped with $AA'= 2AB = 8a$ , $E$ is the midpoint of $(AB)$ and $M$ is the point of $(DD')$ for which $DM = a \left( 1 + \frac{AD}{AC}\right)$. a) Find the position of the point. $F$ on the segment $(AA')$ for which the sum $CF + FM$ has the minimum possible value. b) Taking $F$ as above, compute the measure of the angle of the planes $(D, E, F)$ and $(D, B', C')$. c) Knowing that the straight lines $AC'$ and $FD$ are perpendicular, compute the volume of the parallelepiped $ABCDA'B'C'D'$.


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