Problem 1665

To prove that the quadrilateral formed by the points $K_1, K_2, K_3,$ and $K_4$ is a rhombus if and only if its area is equal to 6 units, we will follow these steps:

1. Identify the points of intersection:
Let the lines $\ell_1$ and $\ell_2$ be given by $y = y_1$ and $y = y_2$ respectively, where $y_1 \neq y_2$. These lines intersect the graph of $f(x) = x^3 + ax^2 + bx + c$ at points $K_1, K_2$ and $K_3, K_4$ respectively.

2. Determine the x-coordinates of the intersection points:
Since $\ell_1$ intersects the graph at $y = y_1$, we solve $x^3 + ax^2 + bx + c = y_1$. Let the roots be $x_1$ and $x_2$. Similarly, for $\ell_2$, solve $x^3 + ax^2 + bx + c = y_2$. Let the roots be $x_3$ and $x_4$.

3. **Form the points $K_1, K_2, K_3,$ and $K_4$:**
The points are $K_1 = (x_1, y_1)$, $K_2 = (x_2, y_1)$, $K_3 = (x_3, y_2)$, and $K_4 = (x_4, y_2)$.

4. Prove the quadrilateral is a rhombus:
For the quadrilateral to be a rhombus, all sides must be of equal length. Calculate the distances between the points:
$$\text{Distance between } K_1 \text{ and } K_2 = |x_2 - x_1|$$
$$\text{Distance between } K_3 \text{ and } K_4 = |x_4 - x_3|$$
$$\text{Distance between } K_1 \text{ and } K_3 = \sqrt{(x_3 - x_1)^2 + (y_2 - y_1)^2}$$
$$\text{Distance between } K_2 \text{ and } K_4 = \sqrt{(x_4 - x_2)^2 + (y_2 - y_1)^2}$$
For the quadrilateral to be a rhombus, we need:
$$|x_2 - x_1| = |x_4 - x_3| \quad \text{and} \quad \sqrt{(x_3 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(x_4 - x_2)^2 + (y_2 - y_1)^2}$$

5. Calculate the area of the rhombus:
The area $A$ of a rhombus can be calculated using the formula:
$$A = \frac{1}{2} \times d_1 \times d_2$$
where $d_1$ and $d_2$ are the lengths of the diagonals. The diagonals can be calculated as:
$$d_1 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
$$d_2 = \sqrt{(x_4 - x_3)^2 + (y_2 - y_1)^2}$$
Given that the area is 6 units, we have:
$$\frac{1}{2} \times d_1 \times d_2 = 6$$
Simplifying, we get:
$$d_1 \times d_2 = 12$$

6. Conclusion:
The quadrilateral formed by $K_1, K_2, K_3,$ and $K_4$ is a rhombus if and only if the area is equal to 6 units, as shown by the conditions derived above.

$\blacksquare$