Thread: Ratio of inscribed and circumscribed triangles

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.

Im getting 1:3 but the answer is 1:4.

Thank You

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: .