Problem 20. The base of the quadrilateral pyramid $M A B C D$ is the parallelogram $A B C D$. Point $K$ bisects edge $D M$. Point $P$ lies on edge $B M$ and $B P: P M=1: 3$. The plane $A P K$ intersects edge $M C$ at point $X$. Find the ratio of segments $M X$ and $X C$.
Solution. For the image of the face $M B C$, we take the right triangle $M B C$ (Fig. 11), and for the direction of projection - the line $A P$. The section $A P X K$ of the pyramid $M A B C D$ is depicted on the drawing as the segment $P K$. To solve the problem, we supplement the rectangular trapezoid $M B C D$ to form a square $M B C H$. Triangles $A M X$ and $H C X$ are similar, and $M A: H C=3: 4$, so $M X: X C=3: 4$.