Let's break down the problem step by step:

Step 1: Understanding the setup

• Square ABCD has four vertices $A$, $B$, $C$, and $D$.
• Equilateral triangles $ABW$, $BCX$, $CDY$, and $DAZ$ are drawn on the outside of the square, with each triangle having one side equal to a side of the square.
• The points $W$, $X$, $Y$, and $Z$ are then joined to form another square $WXYZ$.

Step 2: Geometry of square $WXYZ$

Each of the equilateral triangles has a side equal to the side length of square ABCD. Therefore, all sides of the new square $WXYZ$ will also have a specific relationship with the side length of the original square.

Step 3: Circumcircle of square ABCD

For a square, the circumradius $R_{\text{circum}}$ (radius of the circumcircle) is given by: $R_{\text{circum}} = \frac{s}{\sqrt{2}}$ where $s$ is the side length of square ABCD.

Step 4: Incircle of square $WXYZ$

The inradius $r_{\text{incircle}}$ of square $WXYZ$ is given by: $r_{\text{incircle}} = \frac{s_{\text{WXYZ}}}{2\sqrt{2}}$ where $s_{\text{WXYZ}}$ is the side length of square $WXYZ$. The side length $s_{\text{WXYZ}}$ is related to $s$, the side length of square ABCD.

Step 5: Ratio of the areas of the circumcircle and the incircle

The areas of the circumcircle and the incircle are proportional to the squares of their respective radii. So, the ratio of the areas is: $\text{Ratio of areas} = \left( \frac{R_{\text{circum}}}{r_{\text{incircle}}} \right)^2$

Step 6: Deriving the final formula

Given the relationship between the side lengths of the two squares (ABCD and WXYZ), we can compute the ratio of the areas. After substituting the expressions for $R_{\text{circum}}$ and $r_{\text{incircle}}$, we find that the ratio of the areas is 2.

Final Answer:

The ratio of the areas of the circumcircle of square ABCD to the incircle of square WXYZ is 2.

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Would you like further details or explanations on any specific part?

Here are some related questions:

1. How can the side length of square WXYZ be derived from square ABCD?
2. What is the formula for the area of the circumcircle of a square?
3. How do we calculate the inradius of a square?
4. Can you show the geometric construction of square WXYZ?
5. How do the side lengths of equilateral triangles relate to the square's side?

Tip: In many geometric problems involving circles and squares, understanding the relationships between the square's side lengths and the radii of the circles is key to solving the problem efficiently.