Understanding the Rhombus
A rhombus is a type of polygon that is a quadrilateral with all four sides of equal length. The diagonals of a rhombus bisect each other at right angles, which plays a crucial role in proving the relationship between the sides and diagonals.

Key Definitions
- **Sides (s)**: The length of each side of the rhombus.
- **Diagonals (d1 and d2)**: The lengths of the two diagonals of the rhombus.

Proving the Relationship
To prove that the sum of the squares of the sides of a rhombus equals the sum of the squares of its diagonals, we can use the following steps:
1. **Identify Properties**:
- Let each side of the rhombus be of length \( s \).
- The diagonals intersect at right angles and bisect each other.
2. **Use the Pythagorean Theorem**:
- If \( d1 \) and \( d2 \) are the diagonals, then each half of the diagonals will be \( \frac{d1}{2} \) and \( \frac{d2}{2} \).
- In each triangle formed by the diagonals, apply the Pythagorean theorem:
\[
s^2 = \left( \frac{d1}{2} \right)^2 + \left( \frac{d2}{2} \right)^2
\]
3. **Express in Terms of Diagonals**:
- Therefore,
\[
s^2 = \frac{d1^2}{4} + \frac{d2^2}{4}
\]
- Multiplying through by 4 gives:
\[
4s^2 = d1^2 + d2^2
\]
4. **Conclusion**:
- The sum of the squares of the sides (4s^2) is equal to the sum of the squares of the diagonals (d1^2 + d2^2):
\[
4s^2 = d1^2 + d2^2
\]

Final Result
Thus, we have proven that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.