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: .