Closed-form condition for concyclicity

Question

$A_i(x_i, y_i)$, where $i=0,1,2,3$, are four points such that no three of them are collinear.

Is there a closed form on the condition that they are concyclic?

Answers with $x_0 = y_0 = x_1 (\text { or } y_1) = 0$ are also welcome.

Answer 1

• Construct the circumcircle of $A_0$, $A_1$ and $A_2$. This circle will be unique and well-defined, as it has been stipulated that no three points are collinear. This circle can be found in closed form in a few ways – by the method I described here, for example. Let the centre of this circle be $(R_x,R_y)$ and its radius $r$.
• Now test whether $A_3$ lies on the circle just constructed: $(R_x-x_3)^2+(R_y-y_3)^2=r^2$. The four points are concyclic if and only if this last equation holds.

Note that I have given this as an algorithm rather than a formula, since a full formulaic description would be too unwieldy. Nevertheless, since each step has a closed form, so does the overall algorithm.