1. The problem is to find the center of a circle given its equation or a geometric figure. 2. The center of a circle is the point equidistant from all points on the circle. 3. If the circle is given by the equation $$ (x - h)^2 + (y - k)^2 = r^2 $$, then the center is at the point $$ (h, k) $$. 4. If you have a circle drawn, the center is the point from which all radii extend equally. 5. To find the center from a general circle equation like $$ x^2 + y^2 + Dx + Ey + F = 0 $$, complete the square: - Group x and y terms: $$ (x^2 + Dx) + (y^2 + Ey) = -F $$ - Complete the square for x: $$ x^2 + Dx = (x + \frac{D}{2})^2 - \left(\frac{D}{2}\right)^2 $$ - Complete the square for y: $$ y^2 + Ey = (y + \frac{E}{2})^2 - \left(\frac{E}{2}\right)^2 $$ 6. Substitute back: $$ (x + \frac{D}{2})^2 + (y + \frac{E}{2})^2 = \left(\frac{D}{2}\right)^2 + \left(\frac{E}{2}\right)^2 - F $$ 7. The center is $$ \left(-\frac{D}{2}, -\frac{E}{2}\right) $$. 8. This method works for any circle equation. 9. The radius is the square root of the right side. 10. The center is the point from which all points on the circle are equally distant. Final answer: The center of the circle is $$ \left(-\frac{D}{2}, -\frac{E}{2}\right) $$ if the equation is $$ x^2 + y^2 + Dx + Ey + F = 0 $$ or $$ (h, k) $$ if the equation is in standard form $$ (x - h)^2 + (y - k)^2 = r^2 $$.