The area of an equilateral triangle inscribed in the circle `x^(2)+y^(2)+2gx+2fy+c=0` is

Options

A. `pi(g^(2)+f^(2)-c)^(2)`
B. `1/4(g^(2)+f^(2)=c)`
C. `(3sqrt(3))/4(g^(2)+f^(2)-c)`
D. `(sqrt(3))/4(g^(2)+f^(2)-c)`

Answer

To find the area of an equilateral triangle inscribed in the circle given by the equation \(x^2 + y^2 + 2gx + 2fy + c = 0\), we can follow these steps:

### Step 1: Identify the center and radius of the circle
The equation of the circle can be rewritten in standard form to identify its center and radius. The center of the circle is given by the coordinates \((-g, -f)\) and the radius \(r\) is calculated as:
\[ r = \sqrt{g^2 + f^2 - c} \]

### Step 2: Relate the side length of the triangle to the radius
Let the side length of the equilateral triangle be \(a\). When we draw a perpendicular from the center of the circle to the base of the triangle, we create two right triangles. The perpendicular bisects the base, making each half of the base equal to \(\frac{a}{2}\).

### Step 3: Use trigonometry to find the relationship between \(a\) and \(r\)
In the right triangle formed, we know that the angle at the center corresponding to the base of the triangle is \(60^\circ\). Therefore, the angle at the base is \(30^\circ\). Using the cosine function:
\[ \cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\frac{a}{2}}{r} \]
Since \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), we can set up the equation:
\[ \frac{\sqrt{3}}{2} = \frac{\frac{a}{2}}{r} \]
From this, we can solve for \(a\):
\[ a = \sqrt{3} \cdot r \]

### Step 4: Calculate the area of the equilateral triangle
The area \(A\) of an equilateral triangle with side length \(a\) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \(a = \sqrt{3} \cdot r\) into the area formula:
\[ A = \frac{\sqrt{3}}{4} (\sqrt{3} \cdot r)^2 = \frac{\sqrt{3}}{4} \cdot 3r^2 = \frac{3\sqrt{3}}{4} r^2 \]

### Step 5: Substitute the expression for \(r^2\)
Now we substitute \(r^2 = g^2 + f^2 - c\) into the area formula:
\[ A = \frac{3\sqrt{3}}{4} (g^2 + f^2 - c) \]

### Final Result
Thus, the area of the equilateral triangle inscribed in the circle is:
\[ \boxed{\frac{3\sqrt{3}}{4} (g^2 + f^2 - c)} \]