Two circles `x^(2)+y^(2)+2g_(1)x+2f_(1)y+c_(1)=0` and `x^(2)+y^(2)+2g_(2)x+2f_(2)y+c_(2)=0` intersect at an angle of `120^(@)` then
`(2g_(1)g_(2)+2f_(1)f_(2)-c_(1)-c_(2))^(2)=(g_(1)^(2)+f_(1)^(2)-c_(1)) (g_(2)^(2)+f_(2)^(2)-c_(2))`.
is this statement true or false?

Answer

To determine whether the statement is true or false, we will follow the steps outlined in the video transcript to derive the condition for two circles intersecting at an angle of \(120^\circ\).

### Step-by-Step Solution

1. **Identify the Circles**:
The equations of the circles are given as:
\[ x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0 \quad \text{(Circle 1)} \]
\[ x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0 \quad \text{(Circle 2)} \]

2. **Find the Centers and Radii**:
- The center of Circle 1, \(C_1\), is \((-g_1, -f_1)\) and its radius \(r_1\) is given by:
\[ r_1 = \sqrt{g_1^2 + f_1^2 - c_1} \]
- The center of Circle 2, \(C_2\), is \((-g_2, -f_2)\) and its radius \(r_2\) is given by:
\[ r_2 = \sqrt{g_2^2 + f_2^2 - c_2} \]

3. **Calculate the Distance Between Centers**:
The distance \(d\) between the centers \(C_1\) and \(C_2\) is:
\[ d = \sqrt{(g_2 - g_1)^2 + (f_2 - f_1)^2} \]

4. **Use the Angle of Intersection**:
The angle \(\theta\) between the two circles is given as \(120^\circ\). The cosine of the angle can be expressed as:
\[ \cos(120^\circ) = -\frac{1}{2} \]

5. **Apply the Angle Intersection Formula**:
The formula for the angle of intersection of two circles is:
\[ \cos \theta = \frac{r_1^2 + r_2^2 - d^2}{2r_1r_2} \]
Substituting the values we have:
\[ -\frac{1}{2} = \frac{(g_1^2 + f_1^2 - c_1) + (g_2^2 + f_2^2 - c_2) - d^2}{2 \sqrt{g_1^2 + f_1^2 - c_1} \sqrt{g_2^2 + f_2^2 - c_2}} \]

6. **Substituting for \(d^2\)**:
Substitute \(d^2\) into the equation:
\[ d^2 = (g_2 - g_1)^2 + (f_2 - f_1)^2 \]

7. **Cross-Multiply and Simplify**:
Rearranging gives:
\[ -1 = \frac{(g_1^2 + f_1^2 - c_1) + (g_2^2 + f_2^2 - c_2) - d^2}{\sqrt{g_1^2 + f_1^2 - c_1} \sqrt{g_2^2 + f_2^2 - c_2}} \]
Cross-multiplying leads to:
\[ (g_1^2 + f_1^2 - c_1)(g_2^2 + f_2^2 - c_2) = (2g_1g_2 + 2f_1f_2 - c_1 - c_2)^2 \]

8. **Final Result**:
Thus, we derive the condition:
\[ (2g_1g_2 + 2f_1f_2 - c_1 - c_2)^2 = (g_1^2 + f_1^2 - c_1)(g_2^2 + f_2^2 - c_2) \]
This confirms that the statement is **true**.