$20 \cdot 55$ In an equilateral triangle, the height equals the radius of a circle, and another equilateral triangle is inscribed in this circle. Then the ratio of the perimeters of the two triangles is 
(A) $1: 2$.
(B) $1: 3$.
(C) $1: \sqrt{3}$.
(D) $\sqrt{3}: 2$.
(E) $2: 3$.
(3rd American High School Mathematics Examination, 1952)
[Solution] Given that the height of the equilateral triangle equals the radius and passes through the center $O$ of the circle. The height of the equilateral triangle inscribed in this circle is $\frac{3}{2} R$.
The ratio of the perimeters of similar triangles is equal to their similarity ratio, that is
$$
R: \frac{3}{2} R=2: 3 \text {. }
$$

Therefore, the answer is $(E)$.