Question

show that the points (2,8), (5,7), (-1,-1),(7,3), and (6,0) are concyclic

Answer 1

Taking you at your word, there is a point (x,y) at the same distance d from the above points.

Calculating this distance d for pairs of points and eliminating d gives simultaneous equations in x and y which solve as (2,3) for the center and d = 5 for the radius which also satisfies the fifth point.