Finding the equation of a circle I have the question "Find the equations of the following circles (in some cases more than one circle is possible). 
A circle passes through the points $(1,4)$, $(7,5)$ and $(1,8)$"
I know that to find the equation of a circle I use $(X-a)^2 + (Y-b)^2 = r^2$,
However, I know you do this for one set of points I'm not sure what to do for three set of points and also the radius is not given.
The final answer should be $2X^2 + 2Y^2 -15X - 24Y + 77 = 0$. 

The triangle with vertices at
$$A(1;4),\quad B(7;5),\quad C(1;8) $$
has side lengths $AB=\sqrt{37},\, AC=4,\, BC=3\sqrt{5}$ by the Pythagorean theorem and area 
$$[ABC]=\frac{1}{2}\left|1\cdot 5+7\cdot 8+1\cdot 4-4\cdot 7-5\cdot 1-8\cdot 1\right|=12 $$
by the shoelace formula. It follows that its circumradius is given by
$$R=\frac{abc}{4[ABC]} = \frac{12\sqrt{185}}{48} = \frac{1}{4}\sqrt{185}. $$
The barycentric coordinates of the circumcenter are straightforward to compute through $a^2,b^2,c^2$, hence we know both the centre 
$$ O\left(3+\frac{3}{4};6\right)$$
and the radius of our circumcircle. Its equation readily follows.
