Points A(-2,3),B(-2,5) and C (4,-1) all lie on circle O with center (h,k). find the equation of circle O. how do I do this??

 A(-2,3),B(-2,5) and C (4,-1)

Since all the points will be equidistant from (h,k). Because we have the same x values for two of the points, we can solve this equation for k..... [ we are actually setting r^2  = r^2  ]

(h + 2)^2  + (k - 3)^2   =   ( h + 2)^2 + ( k - 5)^2      subtract the first terms of each side out and we have that

(k -3)^2  = (k - 5)^2        expand

k^2 - 6k + 9  =  k^2 - 10k + 25     subtract k^2 from both sides

-6k + 9  = -10k + 25       add 10k to both sides, subtract 9 from  both sides

4k = 16    divide both sides by 4

k = 4

Now...we can find h  thusly  using the distance formula 

( h + 2)^2   + (4 -3)^2  = (h -4)^2 + (4 + 1)^2     simplify

h^2  + 4h + 4  + 1   =  h^2 - 8h + 16 + 25

12h = 36 

h = 3

So the center is  ( 3, 4)

And the radius can be found thusly

(3 + 2)^2 + 1^2 =  r^2 

26  = r^2

√26  = r

And the equation is

(x - 3)^2 + (y - 4)^2  = 26

Here's the graph with the center and points shown : https://www.desmos.com/calculator/othhfgdve0