51. Dividing a Triangular Cake. Pavel and Gavel must divide a triangular cake between themselves. Gavel set the condition that he would cut his portion with a straight cut, and Pavel agreed on the condition that he would mark a point $P$ in advance, through which the cut must pass. Since the cake has the same thickness everywhere and is uniform in taste, the problem is a planimetric one. The question is as follows: how should Pavel choose the point $P$ to better protect himself from Gavel's gluttony? The second question: what excess will go to Gavel if Pavel successfully solves the first problem, and Gavel then cuts himself the largest possible portion of the cake?

If the shape of the cake depended on Pavel, he could choose a figure with a center of symmetry (i.e., a circle, square, ellipse, etc.), and place $P$ at this center. Then Gavel would have no advantage. However, the interesting question is, what shape of the cake (while maintaining the initial conditions of the division) would be most convenient for Gavel, and what is the largest excess he could ensure for himself by choosing the shape of the cake wisely?
51. First, note that if Pavel chooses point \( P \) at the centroid of the triangle (Fig. 124), then the best division for

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-122.jpg?height=488&width=600&top_left_y=1652&top_left_x=451)

Fig. 124.

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-122.jpg?height=480&width=596&top_left_y=1662&top_left_x=1044)

Fig. 125.

Gavla will be a division where the cut is parallel to one of the sides of the triangle. To prove this fact, it is sufficient to show that the triangle shaded once in Fig. 124 has a smaller area than the triangle shaded twice. The area of the part \( A B C D \) of the cake is \( 5 / 9 \) of the area of the whole cake, so the cake will be divided between Gavla and Pavel in the ratio \( 5: 4 \).

It is easy to verify that choosing the centroid is the best for Pavel, because with any other position of point \( P \), Gavla could cut not only the trapezoid \( A B C D \) but also an additional piece of the cake (Fig. 125).

We leave it to the readers to answer the remaining questions. In any case, it is easy to prove that if the cake has an unusual shape, as shown in Fig. 126, then for any choice of point \( P \), Gavla can cut at least \( 2 / 3 \) of the cake. Finally, we challenge the readers to prove, using the result of problem 50, that for any shape of the cake, Gavla will not cut more than \( 3 / 4 \) of the cake \( * \).

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-123.jpg?height=520&width=576&top_left_y=548&top_left_x=383)

Fig. 126.