The Problem statement is:
An ant leaves the anthill for its morning exercise. It walks 4 feet east and then makes a 160° turn to the right and walks 4 more feet. It then makes another 160° turn to the right and walks 4 more feet. If the ant continues this pattern until it reaches the anthill again, how many turns would it have to take at minimum?
I found an interesting method to solve this problem, which is very similar to the way we solved this problem from IOQM 2020-21: Credits: Allen Career Institute
We can use a similar analogy in this question by assuming two concentric circles, to one of which (smaller circle) the paths followed by the ant are tangent and are equal chords of the other (bigger) circle.
Here AB is 4 and angle formed is $(180-160)=20$:
Then using similar calculations and method we get Total Number of chords = $360/\gcd(160, 360)$.
Just one problem:
I am unable to see a way to prove that all these would be tangent to the inner circle. I first saw the answer and then thought about this method. It fortunately works, but I cannot understand how. Can we generalise this method for any value of the length of path and degree of turn?
If yes, how can you prove it?