カテゴリー: 幾何学

円に内接する4角形の余弦

\[ \cos A=\frac{\left|DA\right|^{2}+\left|AB\right|^{2}-\left|BC\right|^{2}-\left|CD\right|^{2}}{2\left(\left|DA\right|\left|AB\right|+\left|BC\right|\left|CD\right|\right)} \]

トレミーの定理

\[ \left|\overrightarrow{AB}\right|\left|\overrightarrow{CD}\right|+\left|\overrightarrow{BC}\right|\left|\overrightarrow{DA}\right|=\left|\overrightarrow{BD}\right|\left|\overrightarrow{CA}\right| \]

3点を通る円

\[ \det\left(\begin{array}{cccc} x^{2}+y^{2} & x & y & 1\\ x_{1}^{2}+y_{1}^{2} & x_{1} & y_{1} & 1\\ x_{2}^{2}+y_{2}^{2} & x_{2} & y_{2} & 1\\ x_{3}^{2}+y_{3}^{2} & x_{3} & y_{3} & 1 \end{array}\right)=0 \]