From a circle to its numbers and back

You will write three functions. One turns a centre and a radius into the coefficients of the general equation. One reads the centre and the radius from the general equation. One finds the circle through three points. The checker tests them on the circles of the lesson.

The idea. The circle $x^2 + y^2 + Dx + Ey + F = 0$ has centre $(-D/2, -E/2)$ and radius $\sqrt{D^2/4 + E^2/4 - F}$. The centre $(h, k)$ and radius $r$ give $D = -2h$, $E = -2k$ and $F = h^2 + k^2 - r^2$. For three points, the centre lies on two perpendicular bisectors.

Your tasks.

1. Finish general_form(h, k, r) to return (D, E, F).
2. Finish centre_radius(D, E, F) to return (h, k, r).
3. Finish through_three(p1, p2, p3) to return (h, k, r). Put the centre equal in distance from the three points. Subtracting the equations for two pairs of points gives two linear equations in h and k.