How to prove points belong to the same circle

Not sure what the mess above me says, but basically as long as you have an equation for the circle, you can plug the coordinates of your point into the x and y values of the equation and see if it equals out. If you don’t have an equation, you can just make one. Use (f-k)^2+(x-h)^2=r^2 where h and k are the coordinates of the center of the circle and r is the radius. Hope this helps!