17. What is the maximum number of parts into which the plane can be divided by:

a) two triangles;

b) two rectangles;

c) two convex $n$-gons?
17. Let's immediately provide the solution to problem (v), namely, we will prove that the maximum number of parts into which two convex $n$-gons can divide the plane is $2n+2$; from this it will follow that two triangles can divide the plane into no more than 8 parts (problem (a)); see Fig. 54, a), and two rectangles can divide the plane into no more than 10 parts (problem (b)); see Fig. 54, b).

![](https://cdn.mathpix.com/cropped/2024_05_21_15011fcc7946713185f0g-084.jpg?height=227&width=302&top_left_y=254&top_left_x=269)
a)

![](https://cdn.mathpix.com/cropped/2024_05_21_15011fcc7946713185f0g-084.jpg?height=246&width=366&top_left_y=237&top_left_x=609)

b)

Fig. 54.

The parts into which the plane is divided by two convex $n$-gons can be characterized as follows:

1) (one) part located inside both polygons (if it exists);
2) (one) part located outside both polygons;
3) parts located inside one polygon and outside the other.

The problem, therefore, is to determine the maximum possible number of parts of type 3.

Each such part of the plane is a polygon, the smallest number of sides of which is three. On the other hand, each side of one polygon, as a result of intersection with the second (convex!) polygon, can be divided into no more than three parts (Fig. 55, a); therefore, the total number of sides

![](https://cdn.mathpix.com/cropped/2024_05_21_15011fcc7946713185f0g-084.jpg?height=390&width=403&top_left_y=1004&top_left_x=245)
a)

![](https://cdn.mathpix.com/cropped/2024_05_21_15011fcc7946713185f0g-084.jpg?height=374&width=365&top_left_y=1010&top_left_x=628)

b)

Fig. 55.

of all parts of type 3 does not exceed $3n + 3n$. But no segment can be the boundary of two parts of the plane of type 3 (because if on one side of it lies a part of the plane that is external to one polygon and internal to the second, then on the other side there must lie a part of the plane that is either external or internal to both polygons, i.e., a part of the plane of type 1 or 2); therefore, the number of parts of type 3 does not exceed
$\frac{6n}{3} = 2n$. Thus, the total number of parts into which two convex $n$-gons divide the plane does not exceed

$$
1 + 2n + 1 = 2n + 2
$$

The fact that the number of parts can equal $2n + 2$ is demonstrated by the example of two regular $n$-gons with a common center, one of which is obtained from the other by a rotation through an angle not a multiple of $360^\circ / n$ (Fig. 55, b).