An equilibrium triangle is inscribed in a circle and an equilateral triangle

An equilibrium triangle is inscribed in a circle and an equilateral triangle is described near the circle. Find the ratio of the areas of these triangles

Let the radius of the circle be R.

The ABC triangle is inscribed in a circle, R = AC * √3 / 3 cm.

AC = 3 * R / √3 = R * √3 cm.

Then Sас = АС ^ 2 * √3 / 4 = R2 * 3 * √3 / 4 cm2.

Triangle A1B1C1 is described around a circle, then R = A1C1 * √3 / 6 cm.

A1C1 = 6 * R / √3 = 2 * R * √3 cm.

Then Sa1b1c1 = A1C1 ^ 2 * √3 / 4 = R ^ 2 * 12 * √3 / 4 cm2.

Sa1b1s1 / Saabs = (R ^ 2 * 12 * √3 / 4) / (R ^ 2 * 3 * √3 / 4) = 4/1.

Answer: The area ratio is 4/1.