Let's make a construction involving regular polygons:
► First, we begin with a equilateral triangle, with side $\ell_3 = 1;$
► After, we draw a square on the middle point each side of the initial triangle, with side $\ell_4 = \frac{1}{2} = \frac{\ell}{2}.$
Now, the construction continues, taking one of these steps:
► If the regular polygon have an even number of sides $n$ with length $\ell_n$, then we draw two regular polygons with $n + 1$ sides of length $\ell_{n+1} = \frac{\ell_n}{2},$ from the middle point of the extreme segments.
► If the regular polygon have an odd number of sides $n$ with length $\ell_n$, then we draw one regular polygons with $n + 1$ sides of length $\ell_{n+1} = \frac{\ell_n}{2},$ from the middle point of the unique extreme segment in this case.
To clarify the explanation, we will obtain a figure like the one below:
I have two questions about this:
Q1. This figure is inside a circumference with center in the incenter of the initial equilateral triangle? In affirmative case, what is the radius $R$ of the circumference?
Q2. The sequence of the lengths I adopted in the construction is $$\ell_n = \frac{1}{2^{n-3}}, \quad \forall n \ge 3 $$ If I consider other sequence $\ell_n$, when exists a circumference with center in the incenter of the initial equilateral triangle and radius $R$ in which the figure is inside?