We calculate explicitly that the area of the limiting polygon is $\dfrac{4}{7}A$ where $A$ is the area of the original triangle.
Note that on the $(n+1)$-th iteration we cut off twice as many triangles as we did in the $n$-th iteration. Consider the triangles we cut off in the $(n+1)$-th iteration. Each of these has $\dfrac{1}{3}$ the height and $\dfrac{1}{3}$ the base of a triangle cut off in the $n$-th iteration. Hence the ratio of the areas of these triangles is $\dfrac{1}{9}$. Since the total area of the first triangles we cut off is $\dfrac{1}{3}A$, we have that the area of the limiting polygon is
$$A-\sum\limits_{n=0}^\infty2^n\left(\frac{1}{3}A\right)\left(\frac{1}{9}\right)^n=A-\frac{1}{3}A\frac{1}{1-\frac{2}{9}}=\frac{4}{7}A$$
Whether this actually helps us determine the shape of the limiting figure I am not so sure.
This was motivated by Hagen von Eitzen's calculation and Michael's subsequent observation.
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EDIT: Explanation of $\dfrac{1}{3}$ base and $\dfrac{1}{3}$ height statement.
Consider the polygon at a particular vertex $V$ just before making the $n$-th iteration cuts. Choose a particular edge bordering $V$ and let it have length $9x$.
Now take the $n$-th and $(n+1)$-th iteration cuts. In the figure above, the green triangle is one that is cut off in the $n$-th iteration cuts, and the red triangle is one cut off in the $(n+1)$-th iteration cuts.
Because we trisect each time, one side of the green triangle has length $3x$ and one side of the red triangle has length $x$. Call these sides the base of each triangle. Consider now the length of the altitudes corresponding to these bases, letting the altitude of the green triangle be of length $3y$. By similar triangles we find that the altitude of the red triangle has length $y$ (the similar triangles are outlined in black).
Hence the ratio of area of the red triangle to the area of the green triangle is
$$\frac{\frac{1}{2}xy}{\frac{1}{2}(3x)(3y)}=\frac{1}{9}$$
as claimed.
Fun note: we did not use the fact that the triangles in question are isosceles.
The length of a side is $2r sin(\frac{\pi}{n})$ by trigonometry.
The distance is $r(1-cos\frac{\pi}{n})$, which is the difference between the radius and the length of the line from the center of the circle to the center of the side.
Best Answer
Let G and H, I be the midpoints of BC and DE, as O is the center OG, OH, OI are perpendicular to BC, DE, AF and let R be the radius of the circumscribed circle of the equilateral triangle,
You can find OG and OH by pyth. theorem,
$OG=\sqrt{R^2-18^2}$
$OH=\sqrt{R^2-18^2}$
$OI=\sqrt{R^2-18^2}$
by these
$OG=OH=OI=h$
Also by pyth. theorem you can get the lengths of BY, CZ, DZ, XE, XF, AY
$BY=18-\sqrt{12^2-h^2}$
$CZ=18-\sqrt{12^2-h^2}$
$DZ=18-\sqrt{12^2-h^2}$
$XE=18-\sqrt{12^2-h^2}$
$XF=18-\sqrt{12^2-h^2}$
$AY=18-\sqrt{12^2-h^2}$
With above you can find out $\triangle ABY, \triangle DCZ, \triangle XFE$ are equilateral and the side lengths of each other are equal.
By angle chasing you can find that the all angles of the hexagon is equal 120 and it is a regular hexagon.
As OB bisects $\angle ABC$, $\angle ABO=60$ and the one side of the hexagon will be equal to 12 (radius of the inscribed circle)
And the perimeter will be equal to = 12*6=72
Your answer is correct here is the proof for it