Problem 34. In Figure 23, a quadrilateral pyramid $M A B C D$ is shown, with its base being a parallelogram $A B C D$. Points $H$ and $P$ are the midpoints of edges $D M$ and $B C$, respectively. Point $K$ divides edge $B M$ in the ratio $1: 3$, measured from vertex $M$. Construct the point $T$, where the line intersecting line $K P$ intersects line $B D$.
Solution. Let's take the square $ABCD$ as the base of the given pyramid. We consider the line $KP$ as the direction of projection (Fig. 24). Clearly, $FD = MP = \frac{1}{3} PB$.

From similar triangles $TFD$ and $TPB$, it follows that $TD: DB = 1: 2$. Now, we transfer point $T$ to Figure 23.
![](https://cdn.mathpix.com/cropped/2024_05_21_8debcc8c8285c4d79a3dg-121.jpg?height=404&width=1378&top_left_y=2236&top_left_x=337)

![](https://cdn.mathpix.com/cropped/2024_05_21_8debcc8c8285c4d79a3dg-122.jpg?height=474&width=445&top_left_y=251&top_left_x=357)

Fig. 24
![](https://cdn.mathpix.com/cropped/2024_05_21_8debcc8c8285c4d79a3dg-122.jpg?height=802&width=758&top_left_y=256&top_left_x=935)

Fig. 26