Exploring the Number of Polygons Formed by 9 Distinct Points on a Plane
The task of determining the number of polygons that can be formed by 9 distinct points on a plane, given that no three points are collinear, is an interesting problem that involves both combinatorial and geometric reasoning.
Interpreting the Problem
The core of the problem lies in understanding the arrangement of these points and the various polygons that can be formed based on this arrangement. If the 9 points form a convex nonagon (9-sided polygon), then the problem becomes more straightforward. However, if the points are arranged in a non-convex manner, the possibilities can expand significantly.
Convex Polygons
When considering convex polygons, the solution becomes clearer. A convex polygon is a polygon where all the interior angles are less than 180 degrees, and all the vertices (corners) point outwards. For 9 points, the maximum number of convex polygons that can be formed is determined by choosing subsets of these points that form a convex shape.
For instance, if we take a convex nonagon (which is one of the possible convex polygons), there are 8 possible convex heptagons (7-sided polygons) that can be formed by omitting one of the vertices. Similarly, there are 7 hexagons, 6 pentagons, 5 quadrilaterals, and so on, reducing one point at each step. Therefore, the total number of convex polygons that can be formed is 89, as calculated by a previous solution.
The calculation for the number of convex polygons can be derived from the combinatorial principle of choosing subsets:
$$text{Total convex polygons} 8 7 6 5 4 3 2 1 36$$
Non-Convex Polygons
When the points are arranged in a non-convex manner, the number of possible polygons increases significantly. In such arrangements, polygons can be either convex or concave. Convex polygons are straightforward, but concave polygons add complexity.
For example, if we take a specific arrangement of the points where the removal of a single point can result in 5 distinct hexagons (instead of 1), the count for hexagons increases due to the various possible configurations. This is particularly true for pentagons and quadrilaterals, where the number of possible polygons is determined by whether the remaining points form a single convex polygon or multiple concave polygons.
Based on the given example, the total number of non-self-intersecting polygons formed by removing a point from the set of 9 points can be computed as follows:
$$text{Hexagons} 1 6 times 5 31$$ $$text{Pentagons} 57$$ $$text{Quadrilaterals} 51$$ $$text{Triangles} 35$$
The total number of polygons is the sum of all these possibilities:
$$text{Total polygons} 31 57 51 35 174$$
Variable Arrangements
It's important to note that different arrangements of the 9 points can lead to an even higher number of possible polygons. For example, one arrangement might allow for more concave configurations, increasing the total count beyond 174.
The key takeaway from this problem is that while the maximum number of convex polygons is 36, the number of polygons can vary widely depending on the arrangement of the points. This includes considering both convex and concave configurations.
Conclusion
Understanding the number of polygons that can be formed by 9 distinct points on a plane is not limited to a single fixed number but depends heavily on the arrangement of these points. Whether forming convex or concave polygons, the total count can range from 36 to potentially a much larger number with different point configurations.