Ex: Find the circumcentre of the triangle formed by the points (-2,3) (2,-1) and (4,0)

Shift the origin to (-2,3) then the transformed vertices are (0,0) (4,-4) (6,-3)

2x₁      2y₁       -(x₁² +_^y1²)      2x₁

2x₂      2y₂       -(x₂² +y₂²)      2y₂

      _{1        2                  3}

Circumcentre = (₂/₁ , _3/1)   Where ₁=(2x₁)(2y₂) - (2x₂)(2y₁)

In the above example we get

8   -8   -32   8                                 simplify the row if possible the we get    1   -1   -4     1

12  -6  -45   12                                                                                                      4   -2   -15   4

=  {(-1)(-15) - (-2)(-4)/(-2) - (-4) ,  (-4)(4) - (-15)(1)/(-2) - (-4)}

= {7/2 , -1/2}

But we hav to transform bak the equation to origin

Circum centre = {7/2+(-2), -1/2+3)} = {3/2 , 5/2)

Last step can be omitted if the given vertices are in the form of (0,0) (x₁,y₁) (x₂,y₂)