Find the coordinates of the circumcenter of the triangle whose vertices are `(A(5,-1),B(-1,5),` and `C(6,6)dot` Find its radius also.

Explanation

Let the circumcenter be `O(x,y)`. Then
`(OA)^2=(OB)^2=(OC)^2=("Radius")^2=R^2" "(1)`
or `(x-5)^2+(y+1)^2=(x+1)^2+(y-5)^2`
`=(x-6)^2+(y-6)^2`
Taking first two relations, we get
` (x-5)^2+(y+1)^2=(x+1) ^2+(y-5)^2`
or `x=y " "(2)`
Taking last two relations, we get
`(x+1)^2+(y-5)^2=(x-6)^2+(y-6)^2`
or `(x+1)^2+(x-5)^2=(x-6)^2+(x-6)^2` [From (2)]
or `2x^28x+26=2x^-24x+72`
or `x=(23)/(8)`
or Circumcenter `-=(23//8,23//8)`
or `R^2=(x-5)^2+(y+1)^2=(OA)^2`
` =[(23)/(8)-5]^2+[(23)/(8)+1]^2`
`=(17^2)/(64)+(31^2)/(64)=(1250)/(64)`
or Radius `=(25sqrt2)/(8)"units"`.