Construct the perpendicular bisector of each of the sides. The point of intersection of the three bisectors is the circumcenter. It doesn't matter whether the triangle is obtuse or acute -- this works for all triangles.
If your triangle is defined on the coordinate plane, first use the slope formula to determine the slope of the lines containing two of the side segments. Then use the mid-point formulas to determine the mid-points of the same two sides. With the negative reciprocal of the slope and the mid-point, use the point-slope form of the equation of a line to derive the equation of the perpendicular bisector to each of the two sides. Once you have two equations for two of the perpendicular bisectors, solve the system of equations. The solution set of the system of equations will be the ordered pair representing the circumcenter. Again, it doesn't matter whether the triangle is obtuse or not.
