3点を通る円

求める円の方程式を
\[ x^{2}+y^{2}+ax+by+c=0 \] とする。
このとき、3点\(A,B,C\)は円上にあるのでこの方程式を満たし、
\[ \left(\begin{array}{c} x_{1}^{2}+y_{1}^{2}\\ x_{2}^{2}+y_{2}^{2}\\ x_{3}^{2}+y_{3}^{2} \end{array}\right)+\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\left(\begin{array}{c} a\\ b\\ c \end{array}\right)=0 \] となるので、
\begin{align*} \left(\begin{array}{c} a\\ b\\ c \end{array}\right) & =-\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)^{-1}\left(\begin{array}{c} x_{1}^{2}+y_{1}^{2}\\ x_{2}^{2}+y_{2}^{2}\\ x_{3}^{2}+y_{3}^{2} \end{array}\right)\\ & =\left(\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\right)^{-1}\left(\begin{array}{ccc} y_{2}-y_{3} & y_{3}-y_{1} & y_{1}-y_{2}\\ x_{3}-x_{2} & x_{1}-x_{3} & x_{2}-x_{1}\\ x_{2}y_{3}-y_{2}x_{3} & y_{1}x_{3}-x_{1}y_{3} & x_{1}y_{2}-y_{1}x_{2} \end{array}\right)\left(\begin{array}{c} x_{1}^{2}+y_{1}^{2}\\ x_{2}^{2}+y_{2}^{2}\\ x_{3}^{2}+y_{3}^{2} \end{array}\right) \end{align*} となる。
ここで、3点\(A\left(x_{1},y_{1}\right),B\left(x_{2},y_{2}\right),C\left(x_{3},y_{3}\right)\)は同一直線上にないので、
\[ \det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\ne0 \] である。
これより円の方程式は、
\begin{align*} 0 & =x^{2}+y^{2}+ax+by+c\\ & =x^{2}+y^{2}+\left(\begin{array}{ccc} x & y & 1\end{array}\right)\left(\begin{array}{c} a\\ b\\ c \end{array}\right)\\ & =x^{2}+y^{2}-\left(\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\right)^{-1}\left(\begin{array}{ccc} x & y & 1\end{array}\right)\left(\begin{array}{ccc} y_{2}-y_{3} & y_{3}-y_{1} & y_{1}-y_{2}\\ x_{3}-x_{2} & x_{1}-x_{3} & x_{2}-x_{1}\\ x_{2}y_{3}-y_{2}x_{3} & y_{1}x_{3}-x_{1}y_{3} & x_{1}y_{2}-y_{1}x_{2} \end{array}\right)\left(\begin{array}{c} x_{1}^{2}+y_{1}^{2}\\ x_{2}^{2}+y_{2}^{2}\\ x_{3}^{2}+y_{3}^{2} \end{array}\right)\\ & =\left(\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\right)^{-1}\left\{ \det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\left(x^{2}+y^{2}\right)-\left(\begin{array}{ccc} x & y & 1\end{array}\right)\left(\begin{array}{ccc} y_{2}-y_{3} & y_{3}-y_{1} & y_{1}-y_{2}\\ x_{3}-x_{2} & x_{1}-x_{3} & x_{2}-x_{1}\\ x_{2}y_{3}-y_{2}x_{3} & y_{1}x_{3}-x_{1}y_{3} & x_{1}y_{2}-y_{1}x_{2} \end{array}\right)\left(\begin{array}{c} x_{1}^{2}+y_{1}^{2}\\ x_{2}^{2}+y_{2}^{2}\\ x_{3}^{2}+y_{3}^{2} \end{array}\right)\right\} \\ & =\left(\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\right)^{-1}\left\{ \left(x^{2}+y^{2}\right)\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)-\left(\begin{array}{ccc} x & y & 1\end{array}\right)\left(\begin{array}{ccc} y_{2}-y_{3} & y_{3}-y_{1} & y_{1}-y_{2}\\ x_{3}-x_{2} & x_{1}-x_{3} & x_{2}-x_{1}\\ x_{2}y_{3}-y_{2}x_{3} & y_{1}x_{3}-x_{1}y_{3} & x_{1}y_{2}-y_{1}x_{2} \end{array}\right)\left(\begin{array}{c} x_{1}^{2}+y_{1}^{2}\\ x_{2}^{2}+y_{2}^{2}\\ x_{3}^{2}+y_{3}^{2} \end{array}\right)\right\} \\ & =\left(\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\right)^{-1}\left\{ \left(x^{2}+y^{2}\right)\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)-x\left(\begin{array}{ccc} y_{2}-y_{3} & y_{3}-y_{1} & y_{1}-y_{2}\end{array}\right)\left(\begin{array}{c} x_{1}^{2}+y_{1}^{2}\\ x_{2}^{2}+y_{2}^{2}\\ x_{3}^{2}+y_{3}^{2} \end{array}\right)-y\left(\begin{array}{ccc} x_{3}-x_{2} & x_{1}-x_{3} & x_{2}-x_{1}\end{array}\right)\left(\begin{array}{c} x_{1}^{2}+y_{1}^{2}\\ x_{2}^{2}+y_{2}^{2}\\ x_{3}^{2}+y_{3}^{2} \end{array}\right)-\left(\begin{array}{ccc} x_{2}y_{3}-y_{2}x_{3} & y_{1}x_{3}-x_{1}y_{3} & x_{1}y_{2}-y_{1}x_{2}\end{array}\right)\left(\begin{array}{c} x_{1}^{2}+y_{1}^{2}\\ x_{2}^{2}+y_{2}^{2}\\ x_{3}^{2}+y_{3}^{2} \end{array}\right)\right\} \\ & =\left(\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\right)^{-1}\left\{ \left(x^{2}+y^{2}\right)\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)-x\det\left(\begin{array}{ccc} x_{1}^{2}+y_{1}^{2} & y_{1} & 1\\ x_{2}^{2}+y_{2}^{2} & y_{2} & 1\\ x_{3}^{2}+y_{3}^{2} & y_{3} & 1 \end{array}\right)+y\det\left(\begin{array}{ccc} x_{1}^{2}+y_{1}^{2} & x_{1} & 1\\ x_{2}^{2}+y_{2}^{2} & x_{2} & 1\\ x_{3}^{2}+y_{3}^{2} & x_{3} & 1 \end{array}\right)-1\det\left(\begin{array}{ccc} x_{1}^{2}+y_{1}^{2} & x_{1} & y_{1}\\ x_{2}^{2}+y_{2}^{2} & x_{2} & y_{2}\\ x_{3}^{2}+y_{3}^{2} & x_{3} & y_{3} \end{array}\right)\right\} \\ & =\left(\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\right)^{-1}\left\{ \det\left(\begin{array}{cccc} x^{2}+y^{2} & 0 & 0 & 0\\ 0 & x_{1} & y_{1} & 1\\ 0 & x_{2} & y_{2} & 1\\ 0 & x_{3} & y_{3} & 1 \end{array}\right)+\det\left(\begin{array}{cccc} 0 & x & 0 & 0\\ x_{1}^{2}+y_{1}^{2} & 0 & y_{1} & 1\\ x_{2}^{2}+y_{2}^{2} & 0 & y_{2} & 1\\ x_{3}^{2}+y_{3}^{2} & 0 & y_{3} & 1 \end{array}\right)+\det\left(\begin{array}{cccc} 0 & 0 & y & 0\\ x_{1}^{2}+y_{1}^{2} & x_{1} & 0 & 1\\ x_{2}^{2}+y_{2}^{2} & x_{2} & 0 & 1\\ x_{3}^{2}+y_{3}^{2} & x_{3} & 0 & 1 \end{array}\right)+\det\left(\begin{array}{cccc} 0 & 0 & 0 & 1\\ x_{1}^{2}+y_{1}^{2} & x_{1} & y_{1} & 0\\ x_{2}^{2}+y_{2}^{2} & x_{2} & y_{2} & 0\\ x_{3}^{2}+y_{3}^{2} & x_{3} & y_{3} & 0 \end{array}\right)\right\} \\ & =\left(\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\right)^{-1}\left\{ \det\left(\begin{array}{cccc} x^{2}+y^{2} & 0 & 0 & 0\\ 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)+\det\left(\begin{array}{cccc} 0 & x & 0 & 0\\ 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)+\det\left(\begin{array}{cccc} 0 & 0 & y & 0\\ 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)+\det\left(\begin{array}{cccc} 0 & 0 & 0 & 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)\right\} \\ & =\left(\det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\right)^{-1}\det\left(\begin{array}{cccc} 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) \end{align*} となり、両辺に
\[ \det\left(\begin{array}{ccc} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{array}\right)\ne0 \] を掛けて、
\[ \det\left(\begin{array}{cccc} x^{2}+y^{2} & x & y & 1\\ x_{1}^{2}+y_{1}^{2} & x_{1} & y_{1} & 1\\ x_{2}^{2}+y_{2}^{2} & x_{2} & y_{2} & 1\\ x_{3}^{2}+y_{3}^{2} & x_{3} & y_{3} & 1 \end{array}\right)=0 \] となる。
従って題意は成り立つ。