Question 169
The circles $\mathcal{K}_1$ and $\mathcal{K}_2$ in the figure are the circumcircle and incircle of the equilateral triangle $ABC$. A square $DEFG$ is inscribed in the circle $\mathcal{K}_2$ so that the point $D$ lies on the side $AB$. The circles $\mathcal{K}_3$ and $\mathcal{K}_4$ are of the same size and touch each other, each of them also touches two sides of the square $DEFG$. Determine the ratio of the radii of the circles $\mathcal{K}_1$ and $\mathcal{K}_4$.
