https://r2.erweima.ai/mathbot/image/1718561134214h83v2oyl.jpg

两个正方形，求证蓝色和粉色部分面积想等

To verify that the blue and pink areas are equal in the given image, let's perform a geometric analysis based on the visual provided.

### Visual and Geometric Analysis:

1. **Identify the Squares:**
   - There is a larger outer square.
   - There is a smaller, inner square that is rotated and inscribed within the larger square.

2. **Area of the Larger Square:**
   - Let the side length of the larger square be \( a \).
   - The area of the larger square is \( a^2 \).

3. **Area of the Smaller Square:**
   - Let the side length of the smaller square be \( b \).
   - The area of the smaller square is \( b^2 \).

4. **Partitioning of Areas:**
   - The larger square is divided into four triangular regions (two blue and two pink), with the smaller square in the center.

### Proof of Equality:

1. **Symmetry:**
   - The arrangement of the smaller square within the larger square creates symmetrical triangular areas. Each triangular area is composed of parts from the larger square minus the area of the smaller square.

2. **Calculation of Triangular Areas:**
   - Each of the four triangles formed around the smaller square has an area that contributes to either the blue or the pink regions.
   - The triangles adjacent to each other are of equal area because they are formed by the same segments of the larger square.

3. **Equal Partition:**
   - Since the smaller square divides the larger square into four equal triangles, and the triangles are alternatively colored blue and pink, the total area of the blue regions must be equal to the total area of the pink regions.

### Geometric Calculation:

1. **Area of the Larger Square:**
   \[
   A_{\text{large}} = a^2
   \]

2. **Area of the Smaller Square:**
   \[
   A_{\text{small}} = b^2
   \]

3. **Area Outside the Smaller Square:**
   - The area outside the smaller square (which is the total of the blue and pink regions) is:
   \[
   A_{\text{outside}} = A_{\text{large}} - A_{\text{small}} = a^2 - b^2
   \]

4. **Equal Distribution:**
   - The area outside the smaller square is equally divided into blue and pink regions:
   \[
   A_{\text{blue}} = A_{\text{pink}} = \frac{a^2 - b^2}{2}
   \]

### Conclusion:
The areas of the blue and pink regions are indeed equal because they are symmetrical and formed by dividing the same amount of space around the smaller square within the larger square.

Would you like more details or have any specific questions? Here are some additional questions you might consider:

1. How can coordinate geometry help in verifying these areas?
2. What properties of symmetry are useful in geometric proofs?
3. How do we apply the Pythagorean theorem in such problems?
4. Can you explain how the areas of triangles are calculated here?
5. What are some practical applications of such geometric properties?

**Tip:** Drawing additional auxiliary lines or using coordinate geometry can often help in verifying geometric properties and area calculations more precisely.

Learn how to prove the equality of blue and pink areas in inscribed squares through geometric analysis. This problem involves concepts such as area calculations, symmetry in geometric figures, and the formula for the area of a square. Suitable for students in Grades 10-12.

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