Considering bow-tie shapes
I’m thinking about how to count the number of bow-tie shapes formed in a regular 9-gon. It’s like starting with a convex quadrilateral, removing opposite sides, and replacing them with diagonals. I need to figure out how many distinct self-intersecting quadrilaterals (like a bow or kite) can exist by selecting sets of 4 vertices from the nonagon.
Reflecting on bow-tie possibilities
So, if I replace opposite sides AB and CD with diagonals AC and BD, the remaining edges would be BC and DA, which together form a "bow" shape. In a regular 9-gon, each set of 4 vertices can potentially form two possible bow-tie shapes. But wait, symmetry might make them look the same. So, are they genuinely different?
Counting bow-ties in a nonagon
The question is about finding distinct "bows" (self-intersecting quadrilaterals) in a regular 9-gon. For each set of 4 vertices, I could form two bowties, but I need to be careful about degeneracy—where diagonals line up with sides. The quadrilateral must remain convex, and I need to avoid double-counting distinct figures.
Counting distinct "bows"
For each set of 4 vertices (v1, v2, v3, v4), I have two potential "bow" shapes based on which sides I keep. If I consider all possible 4-vertex subsets from the 9-gon, I get 252 possible bows. But I need to check carefully for cases where symmetry might cause two bows to overlap, even if formed from different vertex sets.
Thought for 39s
