Triangle in Circle in Triangle

Rotate the inner triangle $60^\circ$ (or flip it upside down) to see the famous Triforce from the Legend of Zelda series. (Or the first iteration of the Sierpinski triangle) The new figure, ignoring the circle, consists of $4$ congruent triangles. The area of the small triangle is therefore $\boxed{\frac{1}{4}}$ of the large triangle.

If the inner triangle is flipped upside down, it can be simply seen than the outer triangle can contain 4 numbers of the triangles of the size of the inner one

Let the radius of the circle be denoted by R

Let the side length of the large triangle be denoted by M

Let the side length of the small triangle be denoted by N

It is easy to prove that

$M^2 = 12 R^2$ ,

$N^2 = 3 R^2$

The ratio of the two areas $= N^2 : M^2 = 1 : 4 .\square$