Circle through three points

Find the circle through the three points $(0, 0)$, $(4, 0)$, and $(0, 2)$. Write the circle as $x^2 + y^2 + lx + my + n = 0$ and substitute each point. From $(0, 0)$, $n = 0$; from $(4, 0)$, $16 + 4l = 0$, so $l = -4$; from $(0, 2)$, $4 + 2m = 0$, so $m = -2$.

This gives $x^2 + y^2 - 4x - 2y = 0$, and completing the square yields $(x - 2)^2 + (y - 1)^2 = 5$. The center is $(2, 1)$ and the radius is $\sqrt{5}$. The large dots mark the three points and the center $(2, 1)$.