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Let ( h,k) be  the center  of the circle

Since  the square of the radius  =  the square of the radius we can set up this equation

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

(k -1)^2  = (k - 3)^2 

k^2  - 2k + 1  =  k^2  - 6k + 9

-2k + 1 =  -6k + 9

4k  =  8

k  = 2

Then we have that

(h -1)^2 + (k-1)^2  =  ( h - 9)^2  + (k - 2)^2

Subbing in for k we have

(h -1)^2  + (2 -1)^2  = (h -9)^2  + (2-2)^2    simplify

h^2 - 2h + 1  + 1  =  h^2 - 18h + 81

-2h + 2 =  -18h+ 81

16h  =  79

h = 79/16

So  the center is  (79/16 ,2)

And  the  radius can be  found as

(79/16 - 1)^2  + (2 -1)^2  =  r^2

(63/16)^2  + 1  = r^2

4225/256  =  r^2

So the equation is

( x - 79/16)^2 + (y - 2)^2  = 4225/256