Question

A triangle has corners at `(5 ,8 )`, `(2 ,9 )`, and `(7 ,3 )`. What is the area of the triangle's circumscribed circle?

Answer 1

`(x-51/26)²+(y-101/26)²=(sqrt(8845/2)/13)²`

Explanation:

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`