Problem of the Day #10

POTD November 29, 2023
Author: munch

If we draw a regular octagon and all of its diagonals, we find that inside the octagon, there are eight points where exactly 3 diagonals concur, and one point where exactly four diagonals concur. The diagonals divide the octagon into 80 distinct regions. If we do the same for a regular 12-gon, we find that inside the 12-gon, there are 60 points where exactly three diagonals concur, 12 points where exactly four diagonals concur, and one point where exactly six diagonals concur. These are all the points inside the 12-gon where at least three diagonals concur. How many distinct regions are there in the 12-gon?