A line passing through two distinct points $P_1(x_1,y_1),P_2(x_2,y_2)$ can be expressed by $$\det\left| \begin{array}{ccc} x-x_1&y-y_1 \\ x_2-x_1&y_2-y_1 \\ \end{array} \right|=0$$
Since line is set $A=\{a P_1+bP_2|a+b=1\}$ and it corresponds to determinant properties.
A circle passing three(non-colinear) points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ can be expressed by $$\det\left| \begin{array}{ccc} (x-x_1)^2+(y-y_1)^2 & (x-x_1) & (y-y_1) \\ (x_2-x_1)^2+(y_2-y_1)^2 & (x_2-x_1) & (y_2-y_1) \\ (x_3-x_1)^2+(y_3-y_1)^2 & (x_3-x_1) & (y_3-y_1) \end{array} \right|=0$$ or $$\det\left| \begin{array}{ccc} x^2+y^2 & x & y&1 \\ x_1^2+y_1^2 & x_1 & y_1&1 \\ x_2^2+y_2^2 & x_2 & y_2&1 \\ x_3^2+y_3^2 & x_3 & y_3&1 \\ \end{array} \right|=0$$
Please explain simple and neat as possible(like in determinant sense).