Question

how do you find the circumcenter of the following points A (10,13) B (2,9) & C (10,1).

Answer 1

The circumcenter of a triangle is the point where the perpendicular bisectors of the three sides intersect. To find it, you will
take two vertices,
find the equation for the line between them
Find the midpoint of that line (Easy. It is the average of the coordinates.)
Generate the equation for the perpendicular bisector. (You have the point and the slope is the negative of the reciprocal of the slope of the side of the triangle.)
Repeat the process for a different set of two vertices.
Where those two perpendicular bisectors cross is the circumcenter. You could check with the third set of vertices if you had to.
Alternatively, you could draw the triangle and construct the perpendicular bisectors. It is often quicker if you are good at constructions.

Answer 2

The circumcenter is basically the intersection of three perpendicular bisectors, right?

So, find the slope of each side of the triangle. Then find the midpoint of each side.

The perpendicular slope is opposite reciprocal of the original slope. So, set up a y=mx+b type equation. Solve for b. Since you know that the midpoint is on this line, substitute the midpoint coordinates into the equation (x into x, y into y).

If you do this for all three sides, set the three y=mx+b equations equal to each other.

Then, you will be able to find the intersection point, or the circumcenter.