Hello, spred!

An equilateral triangle is inscribed in a circle,

and the circle is inscribed in another equilateral triangle.

Find the ratio of the area of the smaller triangle to that of the larger triangle. Code:

A
D ∆ - - - - - - - * ∆ * - - - - - - - ∆ E
\ * / \ * /
\ * / \ * /
\ * / \ * /
\ / \ /
\ * / \ * /
\ * / ∆ O \ * /
\* / | \ */
\ / | \ /
C ∆ - - - * - - - ∆ B
* D /
\* */
\ * * * /
\ /
\ /
\ /
\ /
\ /
∆
F

The center of the circle is

The inscribed triangle is

The circumscribed triangle is

is the centroid of and

Draw segment

In right triangle

. . Then the altitude of is

Since

. . Then the altitude of is

The ratio of the altitudes is: .

Therefore, the ratio of the area is: .