<s>[INST] Easier method of finding the equation of the circle circumscribing the triangle formed by 3 lines? The equation of the circle circumscribing the triangle formed by the lines $y = 0, y = x$ and $2x + 3y = 10$ is?
I know this can be done by solving two equations at a time and finding the vertex. Then forming 3 different equations to solve for the centre (abscissa, ordinate) and the radius( which are 3 variables). 
I would like to know if there's a shorter method to find the equation of the circle for any 3 random lines given. (It's easier in this particular case as one of the coordinates is $(0,0)$ which makes $c=0$)
 [/INST] Easy to see that $(0,0)$, $(2,2)$ and $(5,0)$ are the vertices of the triangle.
Let $M\left(\frac{5}{2},b\right)$ be the center of the circle.
Thus, $$\left(\frac{5}{2}\right)^2+b^2=\left(\frac{1}{2}\right)^2+(b-2)^2,$$ which gives $$b=-\frac{1}{2}$$ and $$M\left(\frac{5}{2},-\frac{1}{2}\right).$$
Now, for the radius of the circle we obtain:
$$\sqrt{\frac{25}{4}+\frac{1}{4}}=\sqrt{\frac{13}{2}},$$ which gives the answer:
$$\left(x-\frac{5}{2}\right)^2+\left(y+\frac{1}{2}\right)^2=\frac{13}{2}.$$
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