The distance from each of the point to the center has to be the same (radius)
pick two points and calculate the distances and equate them
1,2 and 3,-4
(x-1)^2 + (y-2)^2 = (x-3)^2 + (y+4)^2
x^2 - 2x + 1 + y^2 -4y + 4 = x^2 -6x+9 + y^2+8y + 16
4x - 12y -20 = 0
now pick 3 , -4 and 5, -6
(x-3)^2 + (y+4)^2 = (x-5)^2 + (y+6)^2
x^2 -6x + 9 + y^2 + 8y + 16 = x^2 -10x + 25 + y^2 + 12y +36
4x -4y -36 = 0 two red equations ...solve for x and y multiply THIS one by -1 and add to the one above to get
-8y + 16 = 0 y = 2
then x = 11 center is 11,2