Equation of a circle 3 points

Question

How does one solve the equation of a circle through three given points?...

(81,45) (81,-45) (85,0)

What is the solution to this???

I'm at a loss

Answer 1

Assume the equation to be of the form $(x-x_0)^2 + (y-y_0)^2 = r^2$ for some unknowns $x_0$, $y_0$ and $z_0$.

Plug in those three points for $(x, y)$ and solve for the unknowns.

Answer 2

The general equation of a circle is $$(x-a)^2+(y-b)^2=r^2,$$ with $(a,b)$ being the center of the circle, and $r$ being the radius.

Note that the first 2 points are symmetric about the $x-$axis, telling you that the y coordinate of the center is zero. So you get 2 equations with 2 unknowns $$(81-a)^2+45^2=r^2$$ $$(85-a)^2+0^2=r^2$$ You can subtract one equation from the other to solve for $a$, and then $r$ should be simple from there.