Let $A_1,B_1,C_1,D_1$ be the midpoints of the sides of a convex quadrilateral $ABCD$ and let $A_2, B_2, C_2, D_2$ be the midpoints of the sides of the quadrilateral $A_1B_1C_1D_1$. If $A_2B_2C_2D_2$ is a rectangle with sides $4$ and $6$, then what is the product of the lengths of the diagonals of $ABCD$ ?
1. **Identify the properties of the quadrilateral and its midpoints:**
   - Let \(A_1, B_1, C_1, D_1\) be the midpoints of the sides of the convex quadrilateral \(ABCD\).
   - Let \(A_2, B_2, C_2, D_2\) be the midpoints of the sides of the quadrilateral \(A_1B_1C_1D_1\).

2. **Understand the properties of \(A_2B_2C_2D_2\):**
   - Given that \(A_2B_2C_2D_2\) is a rectangle with sides 4 and 6.

3. **Use the properties of midpoints and parallelograms:**
   - The quadrilateral formed by the midpoints of a quadrilateral is always a parallelogram. Therefore, \(A_1B_1C_1D_1\) is a parallelogram.
   - Since \(A_2B_2C_2D_2\) is a rectangle, \(A_1B_1C_1D_1\) must be a rhombus. This is because the only parallelogram whose midpoints form a rectangle is a rhombus.

4. **Relate the sides of the rectangle to the diagonals of the rhombus:**
   - The diagonals of the rhombus \(A_1B_1C_1D_1\) are the sides of the rectangle \(A_2B_2C_2D_2\).
   - Therefore, the diagonals of \(A_1B_1C_1D_1\) are 4 and 6.

5. **Use the properties of the diagonals of a rhombus:**
   - The diagonals of a rhombus bisect each other at right angles.
   - Let the diagonals of \(A_1B_1C_1D_1\) be \(d_1 = 4\) and \(d_2 = 6\).

6. **Relate the diagonals of the rhombus to the diagonals of the original quadrilateral \(ABCD\):**
   - The diagonals of \(A_1B_1C_1D_1\) are half the lengths of the diagonals of \(ABCD\).
   - Let the diagonals of \(ABCD\) be \(D_1\) and \(D_2\).

7. **Calculate the lengths of the diagonals of \(ABCD\):**
   - Since \(d_1 = \frac{D_1}{2}\) and \(d_2 = \frac{D_2}{2}\), we have:
     \[
     D_1 = 2 \times 4 = 8
     \]
     \[
     D_2 = 2 \times 6 = 12
     \]

8. **Calculate the product of the lengths of the diagonals of \(ABCD\):**
   - The product of the lengths of the diagonals of \(ABCD\) is:
     \[
     D_1 \times D_2 = 8 \times 12 = 96
     \]

The final answer is \(\boxed{96}\).