3点を通る円
3点を通る円
3点\(A\left(x_{1},y_{1}\right),B\left(x_{2},y_{2}\right),C\left(x_{3},y_{3}\right)\)が同一直線上にないとき、この3点を通る円は
\[ x^{2}+y^{2}-\frac{1}{x_{1}y_{2}+y_{1}x_{3}+x_{2}y_{3}-x_{1}y_{3}-y_{1}x_{2}-y_{2}x_{3}}\left(\begin{array}{ccc} x & y & 1\end{array}\right)\left(\begin{array}{ccc} y_{2}-y_{3} & y_{3}-y_{1} & y_{1}-y_{2}\\ x_{3}-x_{2} & x_{1}-x_{3} & x_{2}-x_{1}\\ x_{2}y_{3}-y_{2}x_{3} & y_{1}x_{3}-x_{1}y_{3} & x_{1}y_{2}-y_{1}x_{2} \end{array}\right)\left(\begin{array}{c} x_{1}^{\;2}+y_{1}^{\;2}\\ x_{2}^{\;2}+y_{2}^{\;2}\\ x_{3}^{\;2}+y_{3}^{\;2} \end{array}\right)=0 \] となる。
3点\(A\left(x_{1},y_{1}\right),B\left(x_{2},y_{2}\right),C\left(x_{3},y_{3}\right)\)が同一直線上にないとき、この3点を通る円は
\[ x^{2}+y^{2}-\frac{1}{x_{1}y_{2}+y_{1}x_{3}+x_{2}y_{3}-x_{1}y_{3}-y_{1}x_{2}-y_{2}x_{3}}\left(\begin{array}{ccc} x & y & 1\end{array}\right)\left(\begin{array}{ccc} y_{2}-y_{3} & y_{3}-y_{1} & y_{1}-y_{2}\\ x_{3}-x_{2} & x_{1}-x_{3} & x_{2}-x_{1}\\ x_{2}y_{3}-y_{2}x_{3} & y_{1}x_{3}-x_{1}y_{3} & x_{1}y_{2}-y_{1}x_{2} \end{array}\right)\left(\begin{array}{c} x_{1}^{\;2}+y_{1}^{\;2}\\ x_{2}^{\;2}+y_{2}^{\;2}\\ x_{3}^{\;2}+y_{3}^{\;2} \end{array}\right)=0 \] となる。
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)\\ & =-\frac{1}{x_{1}y_{2}+y_{1}x_{3}+x_{2}y_{3}-x_{1}y_{3}-y_{1}x_{2}-y_{2}x_{3}}\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}-\frac{1}{x_{1}y_{2}+y_{1}x_{3}+x_{2}y_{3}-x_{1}y_{3}-y_{1}x_{2}-y_{2}x_{3}}\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) \end{align*} となる。
\[ x^{2}+y^{2}+ax+by+c=0 \] 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)\\ & =-\frac{1}{x_{1}y_{2}+y_{1}x_{3}+x_{2}y_{3}-x_{1}y_{3}-y_{1}x_{2}-y_{2}x_{3}}\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}-\frac{1}{x_{1}y_{2}+y_{1}x_{3}+x_{2}y_{3}-x_{1}y_{3}-y_{1}x_{2}-y_{2}x_{3}}\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) \end{align*} となる。
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4角形の対辺同士の内積
\[
\overrightarrow{AB}\cdot\overrightarrow{CD}=\frac{1}{2}\left(b^{2}+d^{2}-p^{2}-q^{2}\right)
\]
4角形の対角線と面積の関係
\[
S=\frac{1}{2}\left(\overrightarrow{AC}\times\overrightarrow{DB}\right)
\]
多角形での内接円の半径
\[
r=\frac{S}{s}
\]
円に外接する4角形の面積
\[
S=\sqrt{abcd}\sin\frac{A+C}{2}
\]