how do you find out the center of a circle that circumscribes a triangle?
See the midperpendicular paragraph.
http://www.bymath.com/studyguide/geo/sec/geo7.htm
To find the center of a circle that circumscribes a triangle, you can use the method known as the "circumcenter". The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect.
Here are the steps to find the circumcenter:
1. Identify the three vertices of the triangle.
2. Determine the equation of the perpendicular bisector for each side of the triangle. To find the perpendicular bisector, you need to find the slope of the line containing the side and find the negative reciprocal of that slope. Also, find the midpoint of each side.
3. Solve the system of equations formed by the three perpendicular bisectors. This can typically be done using standard algebraic techniques such as substitution or elimination.
4. The point of intersection of the perpendicular bisectors is the circumcenter of the triangle. This is the center of the circle that circumscribes the triangle.
The link you provided (http://www.bymath.com/studyguide/geo/sec/geo7.htm) explains this process in more detail. To see an example and illustrations, refer to the section titled "Midperpendicular" on that webpage.