I need a general formula that calculates the equidistant locus of three points $(P_x,P_y)$; in terms of the coordinates of the three points $(A_x, A_y), (B_x,B_y), (C_x,C_y)$.
Setting the distances equal yielded nothing for me. Calculating the intersection of the equidistant locus of two pairs of points depended on the x and y coordinates being different for each point, making it impractical to use.
On
In terms of complex variables, $a=A_{x}+iA_{y}$, etc. $$p= \frac{\left| \begin{array}{ccc} a & a\bar{a} & 1 \\ b & b\bar{b} & 1 \\ c & c\bar{c} & 1 \end{array} \right|} {\left| \begin{array}{ccc} a & \bar{a} & 1 \\ b & \bar{b} & 1 \\ c & \bar{c} & 1 \end{array} \right|}$$
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As said, you need to find the coordinates $(X,Y)$ of the center and the radius $R$ of the circle. So, let us write the equations $$(X-x_A)^2+(Y-y_A)^2=R^2 \tag 1$$ $$(X-x_B)^2+(Y-y_B)^2=R^2 \tag 2$$ $$(X-x_C)^2+(Y-y_C)^2=R^2 \tag 3$$ Subtract $(1)$ from $(2)$ and $(1)$ from $(3)$. So, you get $$2(x_A-x_B)X+2(y_A-y_B)Y=(x_A^2+y_A^2)-(x_B^2+y_B^2) \tag 4$$ $$2(x_A-x_C)X+2(y_A-y_C)Y=(x_A^2+y_A^2)-(x_C^2+y_C^2) \tag 5$$ which are simple to solve for $X,Y$. When this is done, use $(1)$ to get $R^2$.
The object you are looking for is called circumcenter. The various formulas for its computation are presented in the Wikipedia article on the subject.