To determine the coordinates of the circumcenter of a triangle, we need to follow a few steps:

Step 1: Review the basic properties

The circumcenter is the point of intersection of the perpendicular bisectors of the sides of the triangle. The coordinates of the circumcenter can be calculated using the formula based on the vertices of the triangle.

Step 2: Formula for Circumcenter Coordinates

If the triangle has vertices at $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$, the circumcenter $O(x, y)$ can be found using the following formulas:

$D = 2 \times \left( x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right)$

Then, the coordinates of the circumcenter are:

$x = \frac{ \left( x_1^2 + y_1^2 \right) (y_2 - y_3) + \left( x_2^2 + y_2^2 \right) (y_3 - y_1) + \left( x_3^2 + y_3^2 \right) (y_1 - y_2) }{D}$

$y = \frac{ \left( x_1^2 + y_1^2 \right) (x_3 - x_2) + \left( x_2^2 + y_2^2 \right) (x_1 - x_3) + \left( x_3^2 + y_3^2 \right) (x_2 - x_1) }{D}$

Step 3: Apply the formula

If you have specific coordinates for the triangle’s vertices, substitute them into the formulas above.

Would you like to proceed with a specific triangle, or would you like me to walk you through an example? Feel free to upload any image if you have one.