Radius of circle given 3 points in polar coordinates

Question

So I've found how to construct a circle passing through 3 points in cartesian coordinates using this website. Is there an easier way of doing this using polar coordinates, or is the best way to solve it by converting the points to cartesian coordinates? The only thing I need to know is if these 3 points could all be fit within a circle of radius r.

Answer 1

Using cartesian coordinates and no matrix calculations, it is quite simple.

$$(x_1-a)^2+(y_1-b)^2=r^2 \tag 1$$ $$(x_2-a)^2+(y_2-b)^2=r^2 \tag 2$$ $$(x_3-a)^2+(y_3-b)^2=r^2 \tag 3$$

Subtract $(1)$ for $(2)$ and $(3)$ to get $$2a(x_1-x_2)+2b(y_1-y_2)=(x_1^2+y_1^2)-(x_2^2+y_2^2) \tag 4$$ $$2a(x_1-x_3)+2b(y_1-y_3)=(x_1^2+y_1^2)-(x_3^2+y_3^2) \tag 5$$

So, two linear equations in $a$ and $b$; easy to solve. When you have them, plug in any of $(1)$, $(2)$ or $(3)$ to get $r^2$ then $r$.