The circumference equation with center in #(a,b)# and radius #r# is given by #(x-a)²+(y-b)²=r²#. Given three non aligned points, #P_1,P_2# and #P_3# an unique circumference passes through them. If the circumference pass through those points, the points must verify the circumference equation.
#P_1->(x_1-a)²+(y_1-b)²=r²#
#P_2->(x_2-a)²+(y_2-b)²=r²#
#P_2->(x_3-a)²+(y_3-b)²=r²#
We have then three equations in the unknowns #(a,b,c)#
They read:
#P_1->89 - 10 a + a^2 - 16 b + b^2 =r^2#
#P_2->85 - 4 a + a^2 - 18 b + b^2 = r^2#
#P_3->58 - 14 a + a^2 - 6 b + b^2 =r^2#
We can easily solve those equations taking instead
#P_1-P_2# and #P_2-P_3# and solving for #(a,b)#
So we obtain the solution:
#a=51/26,b=101/26,r=sqrt(8845/2)/13#
