VIII OM - I - Problem 8

In the rectangular prism $ ABCDA_1B_1C_1D_1 $, the lengths of the edges are given as $ AA_1 = a $, $ AB = b $, $ AD = c $. On the face $ A_1B_1C_1D_1 $, a point $ M $ is chosen at a distance $ p $ from the side $ A_1B_1 $, and at a distance $ q $ from the side $ A_1D_1 $, and a parallelepiped is constructed with the base $ ABCD $ and the lateral edge $ AM $. Calculate the area of the lateral faces of this parallelepiped.
Accepting the notation indicated in figure $5$, we calculate the area of the parallelogram $AMQD$ which is a face of the parallelepiped $ABCDMNPQ$.
The parallelogram $AMQD$ and the rectangle $AKLD$ have a common base $AD$ and height $AK$, thus






But $AK = \sqrt{AA_1^2 + A_1K^2} = \sqrt{a^2 + q^2},\ KL = AD = c$; hence






Similarly, we calculate that the area of $AMNB = b \sqrt{a^2 + p^2}$.